genetic_POPULATIONS.doc
Transcript of genetic_POPULATIONS.doc
![Page 1: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/1.jpg)
J Glob Optim (2007) 37:405–436
![Page 2: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/2.jpg)
DOI 10.1007/s10898-006-9056-6
![Page 3: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/3.jpg)
O R I G I N A L P A P E R
![Page 4: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/4.jpg)
On initial populations of a genetic algorithm
![Page 5: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/5.jpg)
for continuous optimization problems
![Page 6: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/6.jpg)
HeikkiMaaranen · KaisaMiettinen · Antti Penttinen
![Page 7: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/7.jpg)
Received: 7 June 2006 / Accepted: 10 June 2006 /
![Page 8: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/8.jpg)
Published online: 20 July 2006
![Page 9: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/9.jpg)
© Springer Science+Business Media B.V. 2006
![Page 10: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/10.jpg)
Abstract Genetic algorithms are commonly used metaheuristics for global optimi- zation, but there has been very little research done on the generation of their initial population. In this paper, we look for an answer to the question whether the initial population plays a role in the performance of genetic algorithms and if so, how it
![Page 11: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/11.jpg)
should be generated. We show with a simple example that initial populations may
![Page 12: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/12.jpg)
have an effect on the best objective function value found for several generations. Traditionally, initial populations are generated using pseudo random numbers, but
![Page 13: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/13.jpg)
there are many alternative ways. We study the properties of different point genera- tors using four main criteria: the uniform coverage and the genetic diversity of the
![Page 14: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/14.jpg)
points as well as the speed and the usability of the generator. We use the point gen-
![Page 15: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/15.jpg)
erators to generate initial populations for a genetic algorithm and study what effects
![Page 16: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/16.jpg)
the uniform coverage and the genetic diversity have on the convergence and on the final objective function values. For our tests, we have selected one pseudo and one quasi random sequence generator and two spatial point processes: simple sequential
![Page 17: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/17.jpg)
inhibition process and nonaligned systematic sampling. In numerical experiments, we solve a set of 52 continuous test functions from 16 different function families, and
![Page 18: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/18.jpg)
analyze and discuss the results.
![Page 19: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/19.jpg)
Keywords Global optimization · Continuous variables · Evolutionary algorithms ·
![Page 20: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/20.jpg)
Initial population · Random number generation
![Page 21: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/21.jpg)
H. Maaranen
![Page 22: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/22.jpg)
Patria Aviation Oy, Lentokonetehtaantie 3, FI-35600 Halli, Finland
![Page 23: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/23.jpg)
K. Miettinen (B)
![Page 24: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/24.jpg)
Helsinki School of Economics, P.O. Box 1210, FI-00101 Helsinki, Finland
![Page 26: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/26.jpg)
A. Penttinen
![Page 27: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/27.jpg)
Department of Mathematics and Statistics,
![Page 28: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/28.jpg)
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
![Page 29: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/29.jpg)
J Glob Optim (2007) 37:405–436 406
![Page 30: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/30.jpg)
1 Introduction
![Page 31: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/31.jpg)
When solving real optimization problems numerically, the solution process typically
![Page 32: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/32.jpg)
involves the phases of modeling, simulation and optimization. A simulated model of a real life problem is often complex, and the objective function to be minimized may
![Page 33: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/33.jpg)
be nonconvex and have several local minima. Then, global optimization methods are needed to prevent the stagnation to a local minimum. Therefore, in the recent years, there has been a great deal of interest in developing methods for solving global opti- mization problems (see, e.g., [12, 18, 32] and references therein). Here, we consider
![Page 34: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/34.jpg)
global continuous optimization problems.
![Page 35: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/35.jpg)
Genetic algorithms [15, 16, 25] are metaheuristics used for solving problems with
![Page 36: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/36.jpg)
both discrete and continuous variables. The population is the main element of genetic
![Page 37: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/37.jpg)
algorithms, and the genetic operations like crossover and mutation are just instru- ments for manipulating the population so that it evolves towards the final population
![Page 38: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/38.jpg)
including a “close to optimal” solution. The requirements set on the population also
![Page 39: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/39.jpg)
change during the execution of the algorithm.
![Page 40: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/40.jpg)
In the recent years, genetic operators have been developed intensively (see, for
![Page 41: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/41.jpg)
example, [25, 26] and references therein). In addition, most of the theoretical studies involve the tuning or controlling of the parameters of genetic operators and their role
![Page 42: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/42.jpg)
is often considered significantperformance-wise (see, forexample, [11]).However, the role of the initial population, which is the topic of this paper, is widely ignored. Often, the whole area of research is set aside by a statement “generate an initial population,” without implying how it should be done. We show with a simple example that initial
![Page 43: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/43.jpg)
populations may have effects, on the best objective function value found, and these effects may last for several generations. Then, we continue to study whether the tra- ditional way of generating initial populations is recommendable or whether there are other point sets that give faster convergence. Our motivation is to encourage discus-
![Page 44: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/44.jpg)
sion onwhether one should paymore attention to the generation of initial populations.
![Page 45: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/45.jpg)
We concentrate on the casewhere there is no a priori information about the location
![Page 46: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/46.jpg)
of the global minima. Then, initial populations of genetic algorithms are traditionally
![Page 47: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/47.jpg)
generated“randomly.” In practice, genuinerandom (truly independent) points cannot be generated numerically, and instead, pseudo random points (see, for example, [14]) are used, which imitate genuine random points. However, beside [23, 24], there is, up to our knowledge, practically no research done on whether the initial population
![Page 48: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/48.jpg)
should be random.
![Page 49: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/49.jpg)
In [23, 24], quasi random sequences are used to generate initial populations for ge-
![Page 50: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/50.jpg)
netic algorithms. Quasi random points do not imitate random points but are designed
![Page 51: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/51.jpg)
to maximally avoid each other [33]. Quasi random sequences are used in numerical
![Page 52: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/52.jpg)
integration [5, 35, 38], computer simulations [1, 22] and quasi random searches [21, 29, 37] with good success. For example, in [2, 3], modified controlled random searches
![Page 53: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/53.jpg)
and topographical multilevel single linkage are proposed, respectively, which use
![Page 54: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/54.jpg)
quasi random sequences when generating an initial set of solutions. The comparison shows that the proposed methods perform significantly better than the other algo- rithms in the comparison [2, 3]. However, in [2, 3], the influence of the use of quasi random sequences alone cannot be estimated since also other modifications are made
![Page 55: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/55.jpg)
simultaneously to the algorithms.
![Page 56: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/56.jpg)
In this paper, we single out the effect of the different initial population for genetic
![Page 57: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/57.jpg)
algorithms by keeping the rest ofalgorithm identical.We study the influence of the ini- tial population more generally than in [23, 24] and discuss the properties of different
![Page 58: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/58.jpg)
J Glob Optim (2007) 37:405–436
l u ∈Rn
407
i i i ,
i
![Page 59: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/59.jpg)
types of point generators and the effects that different initial populations have on the convergence and the final objective function values. We also collect information and references about different ways of generating initial populations for those who are
![Page 60: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/60.jpg)
interested in alternativeways. For the convenience of the reader,we briefly summarize ways of generating points not widely used in the field ofoptimization. In thenumerical tests, we use a well-established pseudo random number generator, a so-called Nie-
![Page 61: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/61.jpg)
derreiter quasi random sequence generator, which has performed well in our earlier
![Page 62: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/62.jpg)
tests [23], simple sequential inhibition (SSI) process [9] and nonaligned systematic
![Page 63: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/63.jpg)
sampling, which originates from sampling design [34]. Collectively, we call different ways to generate initial populations point generators. The SSI process and the non- aligned systematic sampling are commonly used in statistics where point generators are called spatial point processes [9]. Spatial point processes are used, for example, in the statistical analysis of biological phenomena when simulating a distribution of a
![Page 64: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/64.jpg)
population of plants or animals (see, e.g., [9, 19] and references therein).
![Page 65: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/65.jpg)
Innumerical tests, we fix the geneticalgorithm and its parameter values and change
![Page 66: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/66.jpg)
only initial populations.We use a test suite of 52 test functions from 16 different func- tion families and test whether the differences in best final objective function values
![Page 67: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/67.jpg)
found are statistically significant between the different variants of genetic algorithms. We also study the convergence during the first generations by stopping the algorithm
![Page 68: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/68.jpg)
prematurely after 10 and 20 generations.
![Page 69: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/69.jpg)
The rest of the paper is organized as follows. In Section 2, we show that the initial
![Page 70: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/70.jpg)
population may have an effect on the convergence of a genetic algorithm and give some further motivation for this work. We also shortly present the genetic algorithm used. In Section 3, we give an overview to different point generators that can be used
![Page 71: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/71.jpg)
when generating initial populations, and in Section 4, we discuss and evaluate their speed and usability as well as the coverage and the genetic diversity of the points gen- erated. The numerical results of applying the different variants of genetic algorithms to our test suite are presented and analyzed in Section 5. In Section 6, we discuss the
![Page 72: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/72.jpg)
results, and finally, in Section 7, we conclude the paper and present some directions
![Page 73: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/73.jpg)
for future research.
![Page 74: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/74.jpg)
2 Preliminaries
![Page 75: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/75.jpg)
Population-based genetic algorithms (see, e.g., [15] and references therein) are de-
![Page 76: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/76.jpg)
signed for solving problems that may have several local minima. They are very gen- eral problem solvers, which means that they can be used for solving a wide range of problems. On the other hand, they do not exploit problem-specific information,
![Page 77: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/77.jpg)
which makes them less efficient. Hence, genetic algorithms ought to be used when
![Page 78: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/78.jpg)
problem-specific methods are not available or if a wide range of problems need to be solved with a single algorithm. We consider global optimization problems of the
![Page 79: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/79.jpg)
following form
![Page 80: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/80.jpg)
minimize f (x)
![Page 81: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/81.jpg)
subject to x l ≤ x ≤ xu i = 1, ... , n,
![Page 82: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/82.jpg)
where f : R
n → R is the objective function, x are the decision variables and x , x
![Page 83: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/83.jpg)
are the vectors containing the lower and upper bounds for the decision variables,
![Page 84: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/84.jpg)
respectively.
![Page 85: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/85.jpg)
10
J Glob Optim (2007) 37:405–436
(x /4000 )− (cos(x ) /
408
i=1
2 i
10
i=1
i √
i) i ≤ 100,
i=1
10
1 +i 30 | 2
k=1
k i
k i
2 k , i < 1.
![Page 86: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/86.jpg)
A simple genetic algorithm includes three basic genetic operations: selection,
![Page 87: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/87.jpg)
crossover and mutation. In selection, some solutions from the population are selected
![Page 88: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/88.jpg)
as parents, in crossover the parents are crossbred to produce offspring and in muta- tion the offspring may be altered according to mutation rules. In genetic algorithms, solutions x are called individuals and iterations of an algorithm are called generations.
![Page 89: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/89.jpg)
Many genetic algorithms also employ elitism, which means that a number of the best individuals are copied to the next population. We use a real-coded genetic algorithm that employs tournament selection, heuristic crossover, Michalewicz’s nonuniform
![Page 90: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/90.jpg)
mutation and elitism. For further details of the genetic operations, see [25].
![Page 91: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/91.jpg)
The algorithm used has the following parameters. Population size is the number
![Page 92: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/92.jpg)
of individuals in a population and elitism size is the number of fittest individuals that are copied directly to the next generation. The fitness is evaluated using the objective function (fitness function) value f (x). Tournament size is the number of individuals randomly picked from the whole population for the tournament selection. Crossover rate and mutation rate are probabilities on which the parents are crossbred and off- spring mutated, respectively.Max generations, steps,andtolerance are parameters for the stopping criteria. The algorithm is stopped if a maximum number of generations (max generations) is reached or if there is no change (within the tolerance) in the best
![Page 93: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/93.jpg)
objective function value during the last steps generations.
![Page 94: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/94.jpg)
Genetic algorithms are Markov chains (see, e.g., [31, 46] and references therein).
![Page 95: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/95.jpg)
Such chains are expected to converge to an equilibrium distribution independent of
![Page 96: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/96.jpg)
the initial state. However, in many applications the state space of the chain (possible
![Page 97: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/97.jpg)
point configurations in the feasible region) is extremely large and convergence can be very slow. This poses the question whether the number of the generations used is
![Page 98: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/98.jpg)
sufficient in order to achieve the equilibrium. The initial configuration is one factor in
![Page 99: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/99.jpg)
the speed of convergence. This is our motivation for studying empirically the role of
![Page 100: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/100.jpg)
initial populations.
![Page 101: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/101.jpg)
The difference in the early generationsbecomes important in solvingmany real-life
![Page 102: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/102.jpg)
problems, where the evaluation of the objective functionmay requiretime-consuming simulations and the optimization algorithm may have to be stopped prematurely. We
![Page 103: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/103.jpg)
also expect that the influence of the initial population may carry further, in terms of generations, when solving of the problem requires a large number of function
![Page 104: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/104.jpg)
evaluations.
![Page 105: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/105.jpg)
Figures 1 and 2 illustrate the convergence of a genetic algorithm for the 10-dimen-
![Page 106: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/106.jpg)
sional Griewangk function and 10-dimensional Katsuura function:
![Page 107: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/107.jpg)
Griewangk function : f (x) =
−10 ≤ x
Katsuura function: f (x) =
x − 2 x |−0.1 < x
![Page 108: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/108.jpg)
Each curve illustrates the average convergence for 100 separate runs of a genetic algorithm. The three different curves in Figs. 1 and 2 correspond to runs with differ- ent initial populations generated by using a pseudo random number generator. Here pseudo stands for the case, where the initial pseudo random population is spread out over the whole feasible region. Furthermore, clustered 1 and clustered 2 stand for
![Page 109: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/109.jpg)
cases where the initial population is restricted to a subspace of the feasible region. The
![Page 110: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/110.jpg)
1500
1000
500
J Glob Optim (2007) 37:405–436 409
5 10 15 20 0
50
60
Bes
t obj
ectiv
e fu
nctio
n va
lue
foun
d
10
20
30
40
clustered 1 pseudo clustered 2
45 50 55 60 40 0
2000
Bes
t obj
ectiv
e fu
nctio
n va
lue
foun
d
pseudo
i
![Page 111: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/111.jpg)
Generations
![Page 112: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/112.jpg)
Fig. 1 The convergence of a genetic algorithm for 10D Griewangk function with different initial
![Page 113: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/113.jpg)
populations
![Page 114: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/114.jpg)
clustered 1
clustered 2
![Page 115: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/115.jpg)
Generations
![Page 116: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/116.jpg)
Fig. 2 The convergence of a genetic algorithm for 10D Katsuura function with different initial
![Page 117: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/117.jpg)
populations
![Page 118: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/118.jpg)
subspace is defined by restricting each variable x to the upper and lower80% of their total range for clustered 1 and clustered 2, respectively. For theGriewangk function the
![Page 119: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/119.jpg)
curves merge after 30 generations. The Katsuura function requires more generations
![Page 120: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/120.jpg)
and, therefore, also the curves merge later than for the Griewangk function. (Note
![Page 121: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/121.jpg)
the different scales in Figs. 1 and 2.) For both Griewangk and Katsuura functions, the differences in the best objective function values found is quite significant when the number of generations is small. This indicates that the initial population has an effect
![Page 122: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/122.jpg)
on the convergence of a genetic algorithm.
![Page 123: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/123.jpg)
The clustered populations used in the simple example above were only theoretical.
![Page 124: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/124.jpg)
In practice, such initial populations would hardly be used, unless there were some
![Page 125: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/125.jpg)
J Glob Optim (2007) 37:405–436
Fig. 3 Coverage of the connecting lines with different patterns
410
![Page 126: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/126.jpg)
a priori knowledge about the location of the global minima. The rest of the paper
![Page 127: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/127.jpg)
concentrates on more realistic alternative ways for generating initial populations.
![Page 128: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/128.jpg)
When there is no a priori information available on the location and the number
![Page 129: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/129.jpg)
of local optima, then the initial population of a genetic algorithm should be able to reach as large part of the feasible region as possible by means of crossover. We call this property genetic diversity, and it is related to the independence of points.Another desirable property for an initial population is a good uniform coverage. By a good uniform coverage we mean that the points are well spread out to cover the whole fea- sible region. Points have a gooduniform coverage if they do notform clusters or leave relatively large areas of the feasible region unexplored. A good uniform coverage is desired, because then information is obtained throughout the whole feasible region.
![Page 130: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/130.jpg)
This helps to prevent premature convergence.
![Page 131: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/131.jpg)
Genetic diversity and a gooduniform coverage are in practice conflicting objectives,
![Page 132: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/132.jpg)
because an optimal uniform coverage is often achieved using systematic structures,
![Page 133: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/133.jpg)
which limit the set of possible offspring. For example, it is known that for a closely related problem of sphere packing a triangular grid provides the optimal packing in
![Page 134: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/134.jpg)
a 2-dimensional case (see, e.g., [44]). Hence, if the pattern was repeated, the distance between points would reach its maximum, when using triangular grid. However, the set of possible offspring is limited as illustrated in Fig. 3. This happens also if the
![Page 135: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/135.jpg)
sample points form clusters. Therefore, seemingly random (independent) point sets with no clustering are preferred, since they provide information over the whole fea- sible region and a large part of the feasible region can be reached by the means of crossover. Figure 3 illustrates a rectangular grid, triangular grid, clustered points, and seemingly random points with no clustering. In the figure, the points marked with
![Page 136: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/136.jpg)
J Glob Optim (2007) 37:405–436 411
ences.
paper.
![Page 137: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/137.jpg)
small black dots are parent solutions and the lines running through them illustrate all
![Page 138: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/138.jpg)
the possible locations of offsprings.
![Page 139: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/139.jpg)
3 Generating initial population
![Page 140: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/140.jpg)
Many sub-areas of geneticalgorithms have been studied elaborately, but the selection
![Page 141: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/141.jpg)
of an initial population has been widely ignored.Asmentioned earlier, the traditional
![Page 142: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/142.jpg)
way to generate the initial population is to use pseudo random numbers, and more recently also quasi random sequences have been applied in [23, 24]. Pseudo random numbers imitate genuine random numbers and quasi random sequences are designed
![Page 143: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/143.jpg)
to produce points that maximally avoid each other.
![Page 144: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/144.jpg)
Pseudo random initial populations can be generated in numerous ways. The main
![Page 145: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/145.jpg)
classes are congruential and recursive generators. Common congruential generators
![Page 146: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/146.jpg)
include linear, quadratic, inversive, additive and parallel linear congruential genera- tors [14, 45]. Recursive generators include multiplicative recursive, lagged Fibonacci,
![Page 147: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/147.jpg)
multiply-with-carry-generator, add-with-carry and substract-with-borrow generators [14]. There are also pseudo random vector generators, which produce sequences of
![Page 148: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/148.jpg)
vectors instead of scalars.Examples of those are feedback shift register generator [14]
![Page 149: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/149.jpg)
and SQRT generator [43].
![Page 150: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/150.jpg)
Common quasi random sequence generators include Van der Corput, Hammers-
![Page 151: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/151.jpg)
ley, Halton, Faure, Sobol’ and Niederreiter generators [4, 14, 30]. For the convenience of the reader some examples of both pseudo random number generators and quasi
![Page 152: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/152.jpg)
random sequence generators are included in the Appendix along with further refer-
![Page 153: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/153.jpg)
Pseudo random numbers and quasi random sequences are quite well-known for
![Page 154: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/154.jpg)
researchers in optimization, but there are also other types of point generators. Spatial point processes are commonly used in statistics but they are less well-known in opti-
![Page 155: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/155.jpg)
mization. Therefore, we now shortly describe the spatial point processes used in this
![Page 156: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/156.jpg)
3.1 Spatial point processes
![Page 157: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/157.jpg)
Spatial point processes [9] can be considered to betransformations of pseudorandom point initialization which lead to a good uniform coverage and simultaneously avoid
![Page 158: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/158.jpg)
periodicity in the configuration generated.
![Page 159: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/159.jpg)
Some spatial point processes include a parameter, by which the proportion of the
![Page 160: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/160.jpg)
two conflicting properties, genetic diversity and a good uniform coverage, can be con- trolled. Hence, the same process can be used to generate genetically diverse points or
![Page 161: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/161.jpg)
points with a good uniform coverage or something in between.
![Page 162: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/162.jpg)
There are several types of spatial point processes. For our purposes, it is ade-
![Page 163: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/163.jpg)
quate to differentiate between clustering and inhibition processes. Clustering pro- cesses generate points that simulate populations, where the individuals tend to form clusters. However, we are more interested in the situation where each individual is as far apart as possible from the adjacent individuals resulting in an evenly dis- tributed population without clusters. Therefore, we concentrate on inhibition pro-
![Page 164: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/164.jpg)
cesses, where an individual (a point) is either explicitly prohibited to be located
![Page 165: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/165.jpg)
closer than some predefined minimum distance > 0 to the other individuals, or individuals are kept apart by some implicit means. We describe here two inhibition
![Page 166: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/166.jpg)
S =
J Glob Optim (2007) 37:405–436
j
n k =1 k
= 0, ... , bn −1.
l = (S)mod b
412
End do
n
i,1 j,2 ] ) ,
1. Do i i
Do j i
v −v j
x l, j j )End do
![Page 167: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/167.jpg)
processes: a simple sequential inhibition process and a nonaligned systematic
![Page 168: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/168.jpg)
sampling.
![Page 169: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/169.jpg)
Simple sequential inhibition process: In the simple sequential inhibition (SSI)
![Page 170: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/170.jpg)
process [9], a new individual is accepted to enter the population only if its distance to
![Page 171: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/171.jpg)
all the existing individuals in the population is at least .
![Page 172: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/172.jpg)
In the following pseudo code for the SSI process the parameter population size is
![Page 173: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/173.jpg)
the number of points to be generated, futile is the number of rejected trial points dur- ing the current iteration, Max_futile is the maximum number of rejected trial points in one iteration before terminating the process, and n is the dimension of the space
![Page 174: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/174.jpg)
where the points are generated.
![Page 175: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/175.jpg)
0. Initialize parameters population size,Max_futile and dimension n and set futile = 0
![Page 176: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/176.jpg)
and k = 0.
![Page 177: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/177.jpg)
1. Do until k = = population size or futile==Max_futile
![Page 178: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/178.jpg)
1.1 Generate a trial point.
![Page 179: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/179.jpg)
(Use a pseudo or a quasi random number generator to generate a trial point
![Page 180: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/180.jpg)
to the n-dimensional unit hyper cube.)
![Page 181: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/181.jpg)
1.2 Check whether the trial point is accepted.
![Page 182: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/182.jpg)
If the distance to the existing individuals is larger than , accept the point
![Page 183: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/183.jpg)
into the population, and set k = k + 1 a nd futile= 0.
![Page 184: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/184.jpg)
Else, set futile =futile+1.
![Page 185: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/185.jpg)
The distance between points can becomputed using differentmetrics. If theminimum
![Page 186: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/186.jpg)
distance is defined to be too large, then themaximumnumber of rejected trial points in one iteration is reached and the process is terminated prematurely. In that case, we
![Page 187: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/187.jpg)
supplement the sample with pseudo random points to match the population size.
![Page 188: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/188.jpg)
Nonaligned systematic sampling: In the nonaligned systematic sampling, which orig- inates from sampling design [34], the unit hyper cube is divided into b elementary
![Page 189: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/189.jpg)
intervals (see Appendix) with equal side lengths (i.e., an equally spaced grid). Then one sample point is selected from each elementary interval according to prescribed
![Page 190: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/190.jpg)
rules. Nonaligned systematic sampling uses b · n pseudo random numbers to define
![Page 191: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/191.jpg)
the location of the sample points. In two dimensions, the sample points are generated
![Page 192: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/192.jpg)
using the formula
![Page 193: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/193.jpg)
x = ([(j −1)+r ] , [( i −1)+r
![Page 194: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/194.jpg)
where i, j = 1, ... , b, = 1 / b , a nd r is a b ×n array of pseudo random numbers, see
![Page 195: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/195.jpg)
Fig. 4. In the following pseudo code for an n-dimensional case of the nonaligned sys- tematic sampling, v is a vector of auxiliary integer variables. It is used to determine the correct elementary interval and the correct pseudo random number when calculating
![Page 196: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/196.jpg)
the nonaligned systematic sample points x.
![Page 197: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/197.jpg)
0. Initialize v = 0forj = 0, ... , n −1, and the pseudo random number array .
![Page 198: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/198.jpg)
1.1 Compute sample point x
![Page 199: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/199.jpg)
= 0, ... , n −1 // Compute the jth component of x
![Page 200: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/200.jpg)
i
j = (r +v
![Page 201: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/201.jpg)
Fig. 4 Nonaligned systematic sampling points in two dimensions
1.2 Update v Do j = 0, ... , n −1
j = (v +1)mod b
If ((i)mod b −1) then v
413
j = = bj j
End do End do
2
![Page 202: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/202.jpg)
Note that the number of nonaligned systematic sampling points cannot be chosen
![Page 203: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/203.jpg)
freely, but is determined by the grid size and the dimension. In practice, when a cer-
![Page 204: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/204.jpg)
tain number of sample points is required, we supplement the nonaligned systematic
![Page 205: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/205.jpg)
sample with pseudo random points to match the required number of points.
![Page 206: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/206.jpg)
4 Properties of point generators
![Page 207: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/207.jpg)
In this section, we consider the distribution and the genetic diversity of the points
![Page 208: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/208.jpg)
generated by pseudo random number generators, quasi random sequence generators
![Page 209: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/209.jpg)
and spatial point processes. We also consider the speed and the usability of some specific generators. We are interested in point sets with a relatively small number of points since the number of points generated equals the population size and ranges
![Page 210: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/210.jpg)
typically from tens to a few hundreds.
![Page 211: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/211.jpg)
In the illustrations and numerical examples of point generation, we have selected
![Page 212: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/212.jpg)
good representatives for the different types of point generators. The pseudo random
![Page 213: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/213.jpg)
number generator is a multiplicative linear congruential generator from the well-
![Page 214: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/214.jpg)
established numerical library of the Numerical Algorithms Group Ltd (NAG) and the quasi random sequence generator is the Niederreiter generator [30] that has
![Page 215: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/215.jpg)
proved successful in our earlier tests [23, 24]. The representatives of spatial point
![Page 216: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/216.jpg)
processes are the SSI process and the nonaligned systematic sampling. In the SSI pro-
![Page 217: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/217.jpg)
cess, the trial points are generated using the Niederreiter generator and the distances
![Page 218: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/218.jpg)
are measured using L -metric. The nonaligned systematic sampling is as defined in
![Page 219: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/219.jpg)
Section 3.
![Page 220: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/220.jpg)
The properties of the point generators are described in the following four sub-
![Page 221: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/221.jpg)
sections and the results are summarized and some conclusions are drawn in the fifth
![Page 222: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/222.jpg)
subsection.
![Page 223: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/223.jpg)
as the number of points x ,1
J Glob Optim (2007) 37:405–436 414
n ⊂ Rn 1 N ∈In
n
k N
n N B
A(B ;P)N
n −1
m
![Page 224: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/224.jpg)
4.1 Distribution
![Page 225: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/225.jpg)
Intuitively thinking, it is beneficial if no large areas are left unexplored when sampling individuals for an initial population of a genetic algorithm. The point generators pre-
![Page 226: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/226.jpg)
sented in Section 3 produce point clouds with a degree ofuniform coverage.However, there are differences in how well the points of these sequences are spread out. In opti- mization and in numerical integration, the goodness of the uniform coverage of a
![Page 227: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/227.jpg)
point set is commonly measured by discrepancy or dispersion [30]. Here, we give the
![Page 228: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/228.jpg)
definition for the discrepancy.
![Page 229: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/229.jpg)
Discrepancy Let I be an n-dimensional unit hyper cube, let P be a set consist-
ing of points x , ..., x ,andletB be a nonempty family of Lebesgue-measurable
![Page 230: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/230.jpg)
subintervals of I and B ∈B. Furthermore, let A(B;P)be a counting function defined
![Page 231: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/231.jpg)
≤ k ≤ N, for which xk ∈ B. Then, discrepancy D
![Page 232: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/232.jpg)
with respect to P in I is defined as D (B;P) = sup ∈B − λ ( B ) , w here λ is a
![Page 233: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/233.jpg)
Lebesgue-measure.
![Page 234: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/234.jpg)
Discrepancy is large, when there exists clusters or large unexplored areas. Hence,
![Page 235: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/235.jpg)
we are interested in point sets that have low values for discrepancy. A mathematical relationship between discrepancy and dispersion is given in [30], and it shows that
![Page 236: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/236.jpg)
every low-discrepancy sequence is also a low-dispersion sequence, but not vice versa.
![Page 237: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/237.jpg)
Quasi random sequences are also called low-discrepancy sequences. Their discrep-
![Page 238: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/238.jpg)
ancy value for a large sample size N is of order of magnitude C(logN) N , w here
![Page 239: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/239.jpg)
C is a generator-specific coefficient depending only on the dimension n [27]. This is also a minimum possible discrepancy size for large N (see, e.g., [27]). The bounds for discrepancy, however, are relevant only for a very large number of sample points
![Page 240: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/240.jpg)
as used in numerical integration. When generating only initial populations, we are more interested in the distribution of a small number of points. Some quasi random
![Page 241: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/241.jpg)
sequences have the property that the first b successive points divide evenly on the correspondingelementary intervals (seeAppendix). This indicates that quasirandom sequencesmay have good discrepancy values also forsmall sample sizes. Figure 5 illus-
![Page 242: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/242.jpg)
trates four two-dimensional initial populations with 1,024 individuals generated using the multiplicative linear congruential (pseudo) generator, the Niederreiter generator,
![Page 243: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/243.jpg)
the SSI process and the nonaligned systematic sampling, respectively.
![Page 244: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/244.jpg)
In Fig. 5, we see that pseudo random points form clusters and leave some areas
![Page 245: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/245.jpg)
relatively unexplored, whereas the other samples cover the feasible region quite well.
![Page 246: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/246.jpg)
However, quasi random sequences may sometimes form point patterns, whose dis- tribution properties depend on the number of points generated. Then, at a certain number of sample points the pattern may be unfavorable for optimization purposes. Anexample of this behavior is illustrated in Fig. 6, where again 1,024 points were gen- eratedwith a slightly different configuration. Formore information on the distribution
![Page 247: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/247.jpg)
of different quasi random sequences, we refer to [27].
![Page 248: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/248.jpg)
Next, we consider the distribution of small sample sizes. In optimization with
![Page 249: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/249.jpg)
geneticalgorithms we used a population of 201 individuals. Therefore, we nextempir- ically examine the distribution properties of sets with 201 points generated in four different ways. Henceforth, in tables and figures, we will use abbreviations Pseudo,
![Page 250: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/250.jpg)
Nieder, SSI and Nonalig for the representatives selected from different types of point
![Page 251: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/251.jpg)
generators (see the beginning of this section).
![Page 252: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/252.jpg)
415 J Glob Optim (2007) 37:405–436
Pseudo generator Niederreiter generator
0.5 1 0 0
0.5
1
0.5 1 0 0
1
0.5
0.5 1 0 0
0.5
1
0.5 1 0 0
0.5
1
![Page 253: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/253.jpg)
SSI process
Nonaligned systematic sampling
![Page 254: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/254.jpg)
Fig. 5 Point sets of 1,024 points with different generators
![Page 255: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/255.jpg)
We consider 2 and 10-dimensional cases to show how the distribution properties
![Page 256: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/256.jpg)
may change with the dimension. First, we use an empirical empty space statistic to examine how well the sample points cover the feasible region [34]. Empty space
![Page 257: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/257.jpg)
statistic function ess is defined as
![Page 258: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/258.jpg)
ess(r) = 1 −Pr(B(x, r) is empty),
![Page 259: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/259.jpg)
where x is a randomly chosen point, B(x, r) is an x-centered ball with radius r and Pr denotes the probability. Hence, assume there is a point set S and further assume there is a ball B with radius r and whose center point is randomly chosen from the feasible region (in our case, the unit hyper cube). Then, for each radius r, the empty space statistic function tells what the probability is that the ball B is empty, that is, the
![Page 260: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/260.jpg)
intersection of the ball B and the point set S is empty.
![Page 261: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/261.jpg)
J Glob Optim (2007) 37:405–436
Fig. 6 Another example of 1,024 points generated with the Niederreiter generator
sets.
1
0.5
0
Niederreiter generator
0.5
1
1 To experimentally define the empty space statistic functions for each point set con-
0
1
results.
![Page 262: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/262.jpg)
taining 201 points, we generated 10,000 auxiliary pseudo random points on a unit hyper cube and for each of the 10,000 auxiliary points calculated the maximal radius
![Page 263: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/263.jpg)
r so that B(x, r) is empty, that is, does not contain any of the 201 points under consid- eration. We did this in both 2- and 10-dimensional cases for the four different point
![Page 264: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/264.jpg)
The empirical empty space statistic functions are illustrated in Fig. 7. For us, the
![Page 265: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/265.jpg)
important property of the empty space statistic function is the steepness of the curve
![Page 266: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/266.jpg)
(the steeper the better). If the curve is steep, it indicates that if a random ball with radius r is selected from the unit hyper cube, then whether the ball is empty depends mainly on the radius—not the location of the ball. If, on the other hand, the curve is gentle, the emptiness of the ball depends strongly on the location, hence the point set
![Page 267: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/267.jpg)
is not evenly distributed.
![Page 268: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/268.jpg)
The empiricalempty space statistic functions show that in two dimensions, the non-
![Page 269: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/269.jpg)
aligned systematic sampling and the simple sequential inhibition (SSI) process provide
![Page 270: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/270.jpg)
the best coverages closely followed by the Niederreiter quasi random sequence gen- erator. The pseudo random initial population has clearly the worst coverage, which
![Page 271: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/271.jpg)
confirms the findings earlier shown in Fig. 5. However, in ten dimensions the differ-
![Page 272: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/272.jpg)
ences diminish, because the populations become sparse. The only notable difference is that for the population generated with the SSI process the curve of the empirical
![Page 273: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/273.jpg)
empty space statistic function is now below the other curves with the small values of
![Page 274: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/274.jpg)
r but rises more steeply in the end. This indicates better coverage for the SSI process.
![Page 275: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/275.jpg)
4.2 Genetic diversity
![Page 276: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/276.jpg)
The population of a genetic algorithm evolves largely by crossovers and mutations. In our implementation, the most dominant genetic operator is crossover, since it usually changes the solutions most. We define genetic diversity as a property by which the
![Page 277: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/277.jpg)
Here, we naturally used a different pseudo random number generator than when generating the
![Page 278: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/278.jpg)
initial population. To make sure that the use of pseudo random numbers did not bias the results we also computed the empty space statistic functions using a grid of 10,000 points, which led to similar
![Page 279: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/279.jpg)
J Glob Optim (2007) 37:405–436 417
0.05 0.1 0.15 0.2
ess
(r)
2D
0 0
0.2
0.4
0.6
0.8
1
Pseudo Nieder SSI Nonalig
0.2 0.4 0.6 0.8 1 1.2 1.4 0 0
0.2
0.4
0.6
0.8
1
ess
(r)
10D
Pseudo Nieder SSI Nonalig
n 1 n ,
n n
![Page 280: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/280.jpg)
Radius r
Radius r
![Page 281: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/281.jpg)
Fig. 7 Empirical empty space statistic functions
![Page 282: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/282.jpg)
genetic algorithm is able to reach as large a part of the feasible region as possible by
![Page 283: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/283.jpg)
means of crossover. Genetically more diverse initial populations are preferable.
![Page 284: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/284.jpg)
Next,we concentrate on the independence of points,which is onemain criterion for
![Page 285: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/285.jpg)
genetic diversity.Asmentioned earlier, truly independent points cannot be generated
![Page 286: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/286.jpg)
algorithmically. The pseudorandomnumbers try to imitate independentnumbers, yet they are deterministically generated by an algorithm. For example, it has been shown that the pseudo random points generated by a linear congruential generator lie on a simple lattice (see, e.g., [14] and references therein). It is also known that bad choices for the coefficients of a linear congruential generator may result in a very bad lattice structure (see, e.g., [14]). The choices for the coefficients play a crucial role for all the
![Page 287: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/287.jpg)
pseudo random number generators.
![Page 288: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/288.jpg)
Also quasi random sequences are generated algorithmically. However, the idea of
![Page 289: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/289.jpg)
quasi random points is not to imitate random points but to try to cover the whole
![Page 290: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/290.jpg)
search space yet maintaining a certain degree of randomness. These initial settings speak for a better genetic diversity for pseudo random initial populations. In empiri-
![Page 291: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/291.jpg)
cal tests, when comparing pseudo and quasi initial populations, we can study plots of 2-dimensional populations or plots of n-dimensional populations that are projected
![Page 292: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/292.jpg)
to two dimensions. In Figs. 5 and 6 we can notice that the Niederreiter generator
![Page 293: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/293.jpg)
produces points that have a more distinguishable pattern whereas pseudo random
![Page 294: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/294.jpg)
points have a pattern that seems more random.
![Page 295: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/295.jpg)
We studied the independence of point sets of 201 2-dimensional points using Rip-
![Page 296: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/296.jpg)
ley’s K-function and L-function [34]. The K-function is defined as follows: λK(r) is the expected number of further points within a ball with radius r and with a center at a randomly chosen point ( λ is the expected number of points in a unit hyper cube).
![Page 297: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/297.jpg)
L-function is obtained from the K-function through transformation
![Page 298: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/298.jpg)
L(r) = (K(r) / )
![Page 299: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/299.jpg)
where is the volume of the unit ball in R . Figure 8 illustrates the L-functions for the pseudo random points, the Niederreiter quasi random points, points from the SSI process and the nonaligned systematic sampling points. For independent points
![Page 300: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/300.jpg)
J Glob Optim (2007) 37:405–436
Four L−functions
418
0.05 0.1 0.15 0.2 0.25 0 0
0.05
0.1
0.15
0.2
0.25
Distance r
L fu
nctio
n
Pseudo Nieder SSI Nonalig
we get 2
![Page 301: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/301.jpg)
Fig. 8 L-functions for
![Page 302: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/302.jpg)
different initial populations
![Page 303: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/303.jpg)
the L-function is a straight line L(r)=r. We can see that pseudo random points imi- tate well independent points, whereas Niederreiter quasi random points and the two
![Page 304: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/304.jpg)
spatial point processes lack the points falling close to each other. Based on these
![Page 305: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/305.jpg)
observations, we conclude that the pseudo random generator produces genetically
![Page 306: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/306.jpg)
more diverse points than the other point generators.
![Page 307: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/307.jpg)
The points generated by the SSI process are essentially pseudo random points.
![Page 308: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/308.jpg)
However, the trial points falling too close to each other are not accepted to the popu- lation and this affects the genetic diversity. The genetic diversity versus a gooduniform coverage can be controlled using theminimum distanceparameter . In what follows, we use the term probabilistic maximal minimum distance (PMMD) and we should find the value of PMMD so that a desired number of points can fit in the feasible region with a given probability. In the 2-dimensional example in Fig. 8, we can see
![Page 309: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/309.jpg)
that, except for small distances, the points of the SSI process fall on a straight line.
![Page 310: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/310.jpg)
The nonaligned systematic sampling uses only n ·b pseudo random points, where
![Page 311: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/311.jpg)
n is the dimension and 1/b is the side length of the elementary interval (the distance
![Page 312: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/312.jpg)
between two adjacent grid points). Therefore, there is not much genetic diversity in nonaligned systematic sample points. In Fig. 8, we see this as an oscillating behavior of the L-function. Furthermore, the number of nonaligned systematic sample points increases quickly with the dimension if the grid size is kept constant. For example, in
![Page 313: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/313.jpg)
10 or 20-dimensional cases, even if we use only two grid points along each dimension,
![Page 314: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/314.jpg)
10 = 1, 024 and 220 = 1, 048, 576 sample points, respectively. When using a
![Page 315: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/315.jpg)
population size of 201 individuals, we get in practice only one genuinesystematicsam-
![Page 316: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/316.jpg)
ple point in 8 or more dimensions (the rest are supplemented pseudo random points). This means that, in large dimensions, the nonaligned systematic sampling reduces to
![Page 317: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/317.jpg)
pseudo random sampling.
![Page 318: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/318.jpg)
4.3 Speed
![Page 319: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/319.jpg)
For some applications the speed of the generator may be of importance. The CPU times for the tested generators and the different numbers of points generated in dimensions 2, 5, 10, 20, 50 are shown in Table 1. The test runs were performed on
![Page 320: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/320.jpg)
J Glob Optim (2007) 37:405–436 419
Dim Points SSI Nonalig
2 201 0.00 0.00 0.06 0.00 4 0.00 17.60 0.02 5 0.02 1774.0 0.23
5 201 0.00 0.00 0.2 0.00 4 0.00 17.84 0.06 5 0.03 1760.0 0.67
10 201 0.00 0.00 0.23 0.00 10 10 4 0.03 0.01 17.71 0.04 10 10 5 0.28 0.05 1773.0 0.98 20 201 0.00 0.01 0.73 0.00 20 10 4 0.06 0.01 10.78 0.05 20 10 5 0.55 0.09 21.33 0.46 50 201 0.01 0.00 1.54 0.01 50 10 4 0.14 0.02 44.50 0.12 50 10 5 1.38 0.2 45.68 1.2
and 50.
![Page 321: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/321.jpg)
Tab le 1 CPU times in seconds
Pseudo Nieder
![Page 322: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/322.jpg)
2 1 00.01
![Page 323: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/323.jpg)
2 1 00.06
![Page 324: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/324.jpg)
5 1 00.02
![Page 325: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/325.jpg)
5 1 00.14
![Page 326: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/326.jpg)
an HP9000/J5600 computer, and the CPU times were obtained using NAG routine X05BAF, which returns the CPU time with the accuracy of 0.01 s. For this reason, differences betweensome generators become evident only when using a largenumber
![Page 327: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/327.jpg)
of points. The columnDim in Table 1 indicates thedimensions of the points generated
![Page 328: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/328.jpg)
and the column Points lists the number of points generated. The abbreviations for the
![Page 329: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/329.jpg)
generators are the same as earlier, and the CPU times are given in seconds.
![Page 330: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/330.jpg)
In Table 1, we see that the SSI process is by far the slowest resulting from the
![Page 331: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/331.jpg)
implementation which emphasizes the good coverage at the expense of speed. Our implementation allows a maximum of 2,000 futile trial points at each iteration before
![Page 332: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/332.jpg)
terminating the process. The speed (and the distribution of points) for the SSI process depends strongly on the probabilistic maximal minimum distance PMMD. If PMMD
![Page 333: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/333.jpg)
is selectedmaximally large (i.e., PMMD is as large as possible yet so that the algorithm does not have to supplement the population with pseudo random points, see Section 3.1), then a good coverage of the points is emphasized and the number of futile tries
![Page 334: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/334.jpg)
increases slowing down the process. Our implementation of the SSI process auto-
![Page 335: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/335.jpg)
matically estimates the maximally large PMMD. The values in Table 1 indicate that
![Page 336: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/336.jpg)
the estimate for PMMD is probably not good (distribution-wise) for dimensions 20
![Page 337: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/337.jpg)
Table 1 shows that the implementation of the Niederreiter quasi random number
![Page 338: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/338.jpg)
generator is the fastest followed closely by the pseudo random number generator and the nonaligned systematic sampling process. In optimization, where only a relatively
![Page 339: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/339.jpg)
small number of points is generated, there is no practical difference between the speed of these three generators. For pseudo and quasi random number generators
![Page 340: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/340.jpg)
the speed seems to be directly proportional to the number of points generated. We can see that, compared to other generators, the speed of the nonaligned systematic
![Page 341: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/341.jpg)
sampling grows for largerdimensions. This can be explained by different ratios of gen- uine nonaligned sampling points. As mentioned earlier, the population of nonaligned
![Page 342: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/342.jpg)
systematic sampling points is supplemented by pseudo random points to match the
![Page 343: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/343.jpg)
required population size. Hence, for example, in 5 dimensions there are 78% and
![Page 344: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/344.jpg)
100% of genuine nonaligned systematic sampling points in sets of 10,000 and 100,000
![Page 345: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/345.jpg)
J Glob Optim (2007) 37:405–436 420
20 6
here.
![Page 346: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/346.jpg)
sample points, respectively, whereas in 10 dimensions the respective values are only 10% and59%. In 20 dimensions, all except onesample point are already random even
![Page 347: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/347.jpg)
if the sample size was as large as one million (2 >10 ).
![Page 348: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/348.jpg)
4.4 Usability
![Page 349: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/349.jpg)
In this section, we discuss the usability of different number generators. Some fea-
![Page 350: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/350.jpg)
tures affecting usability have already been mentioned earlier and are summarized
![Page 351: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/351.jpg)
Pseudo random number generators are by far the most commonly used ones in
![Page 352: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/352.jpg)
optimization. The greatest advantage of pseudo random number generators concern- ing usability is that there exist implementations that are well-established, well-tested
![Page 353: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/353.jpg)
and easily available.However, a goodrandom-like behavior is not amatter-of-course,
![Page 354: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/354.jpg)
but requires a careful choice of parameters, that is, coefficients and seeds (or initial sequences). As earlier noted for linear congruential generators, wrong choices for parameter values may cause the points in a sequence to be badly distributed. This is true for other generators as well. Many of the pseudo random number generators are easy to implement, but the analysis of the distribution is difficult, especially for an arbitrary seed (or arbitrary initial sequences). According to [14], only well-tested pseudo random generators should be used. These generators often use fixed coeffi- cients and sometimeslimit the choice of seeds (or initial sequences). This is one way to
![Page 355: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/355.jpg)
secure good distribution properties, assuming those values are tested and approved.
![Page 356: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/356.jpg)
Contrary to pseudo random number generators, quasi random number generators
![Page 357: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/357.jpg)
are neither so well-established, well-tested nor easily available. On the other hand, quasi random number generators having solid theoretical properties do not need so
![Page 358: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/358.jpg)
much numerical testing. The use of available quasi generators is easy and the quasi
![Page 359: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/359.jpg)
random points are guaranteed to have some advantageous properties as described in the Appendix. Quasi generators provide the same sequences on different runs of the
![Page 360: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/360.jpg)
generator and, hence, they do not require a seed or an initial sequence from the user.
![Page 361: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/361.jpg)
Since quasi generators take into consideration the location of the previous points, they also expect that the sequence is startedfrom the beginning. These two properties make quasi generators easier to use, because the user only has to give the number of points to be generated and the dimension as parameters. At the same time, these properties may become a disadvantage if the user for some reason wants to obtain
![Page 362: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/362.jpg)
different sequences in the same dimension. On this occasion, the easiest way out might be to generate points in higher dimensions and then projectthem to the desired
![Page 363: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/363.jpg)
dimension, but this may have an effect on the distribution properties.
![Page 364: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/364.jpg)
The SSI process is not commonly available for an n-dimensional case, but the
![Page 365: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/365.jpg)
implementation is straightforward given that the user provides the PMMD value.
![Page 366: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/366.jpg)
However, it gets more complicated if the user provides only the number of points to
![Page 367: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/367.jpg)
be generated and theoptimalPMMD(with respect to the coverage)must beestimated
![Page 368: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/368.jpg)
automatically. This issue is discussed in more detail in Section 6. An interesting prop- erty of the SSI process is that the proportion of the two conflicting objectives, that is,
![Page 369: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/369.jpg)
the genetic diversity and the good uniform coverage, can both be controlled using the
![Page 370: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/370.jpg)
probabilistic maximal minimum distance parameter PMMD. Note that, if desired, the SSI process provides different points on every run just like pseudo random number
![Page 371: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/371.jpg)
generators.
![Page 372: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/372.jpg)
The nonaligned systematic sampling processes are not commonly available in
![Page 373: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/373.jpg)
n-dimensions, but they are easy to implement and to use. However, when
![Page 374: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/374.jpg)
J Glob Optim (2007) 37:405–436 421
Coverage + ++ +++ +++/+ +++ ++ ++ +
Speed +++ +++ + ++/+++ Usability +++ +++ ++ +++/+
![Page 375: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/375.jpg)
generating a relatively small number of points, it is practical to use the systematic
![Page 376: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/376.jpg)
sampling process only in small dimensions. Otherwise, the proportion of genuine
![Page 377: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/377.jpg)
nonaligned systematic sample points is small. This is a strong limitation in the area of
![Page 378: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/378.jpg)
optimization.
![Page 379: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/379.jpg)
4.5 Summary of features
![Page 380: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/380.jpg)
An ideal generator for our purposes should generate well-distributed, genetically diverse points in n-dimensions, and it should be relatively fast and easy to use. In Table 2, we summarize the evaluation made in this section. The evaluation of differ- ent point generators has been marked using plus signs (+): the more plus signs the better the score. In case of a notable difference in how well a generator works for
![Page 381: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/381.jpg)
problems with small and large dimensions (here small denotes less than, say, five), the occasions are scored separately (small/large). For example, the nonaligned systematic sampling scores +++/+ in coverage,whichmeans that itworkswell insmall dimensions
![Page 382: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/382.jpg)
and poorly in large dimensions.
![Page 383: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/383.jpg)
Note that only coverage and genetic diversitymay have a direct influence on objec-
![Page 384: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/384.jpg)
tive function values inoptimization since they are the properties of the points whereas speed and usability are properties of the generators. Considering just the properties
![Page 385: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/385.jpg)
of the points we notice in Table 2 that pseudo random points have good genetic diver- sity, but the worst coverage, and the SSI process produces points with good coverage, but only average genetic diversity. The properties of the Niederreiter quasi random points settle somewhere between pseudo and the SSI process. In the further analysis, we pay less attention to the nonaligned systematic sampling since it is not applica- ble for problems with several variables. To find out the effects of the coverage and genetic diversity, we concentrate on the pseudo random number generator and the
![Page 386: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/386.jpg)
SSI process.
![Page 387: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/387.jpg)
5 Experimentations
![Page 388: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/388.jpg)
We test the influence of initial populations computationally by using different point
![Page 389: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/389.jpg)
generators and a largenumber of difficult test problems from the literature. The influ-
![Page 390: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/390.jpg)
ence of the different initial populations is analyzed after 10 and 20 generations and after the execution of the whole algorithm. The genetic algorithm used is presented
![Page 391: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/391.jpg)
in Section 2.
![Page 392: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/392.jpg)
5.1 Test settings
![Page 393: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/393.jpg)
The test runs wereperformed on anHP9000/J5600computer.We solved a test suite of
![Page 394: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/394.jpg)
52problems using geneticalgorithms described in Section 2 with theparameter values
![Page 395: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/395.jpg)
Tab le 2 Summary of generator properties
Properties (small/large) Pseudo Nieder SSI Nonalig
![Page 396: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/396.jpg)
Genetic diversity
![Page 397: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/397.jpg)
J Glob Optim (2007) 37:405–436 422
Value
201 21 3 0.8 0.1
Steps 100 1 0 −7
# Name Ref.
1–3 2, 5, 10 [0, π] [13] 4–7 Rastrigin [13] 8–11 Schwefel [13] 12 2 [13] 13–17 [13] 18–22 [13] 23 Easom 2 [13] 24 Levy 4 [−10, 10] [40] 25–27 Levy 5, 6, 7 [−5, 5] [40] 28 P8 3 [−10, 10] [7] 29 P16 5 [−5, 5] [7] 30 Hansen 2 [−10, 10] [40] 31–34 Corona 4, 6, 10, 20 [10] 35–38 Katsuura 4, 6, 10, 20 [−1, 1] [10] 39–42 5, 5, 10, 10 [0, 10] [41, 13] 43–46 3, 4, 10, 20 [−20, 20] [42] 47–49 Shekel 4, 4, 4 [0, 10] [40] 50–52 2, 5, 10 [0, π] [41]
![Page 398: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/398.jpg)
Tab le 3 Parameter values for genetic algorithms
Parameter
![Page 399: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/399.jpg)
Population size
![Page 400: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/400.jpg)
Elitism size
![Page 401: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/401.jpg)
Tournament size
![Page 402: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/402.jpg)
Crossover rate
![Page 403: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/403.jpg)
Mutation rate
![Page 404: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/404.jpg)
Max generations 10, 20 and 2000
![Page 405: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/405.jpg)
To le ran ce
![Page 406: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/406.jpg)
given in Table 3. The only difference between the variants of a genetic algorithm used was the way the initial population was generated. Each problem was solved 10 times with each algorithm, when the algorithms were let run until the stopping criteria were
![Page 407: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/407.jpg)
satisfied, and 100times when thealgorithms were stopped prematurely after 10 and 20 iterations. In each run, we recorded the best objective function value in the last pop- ulation and study them in what follows. The names of the function families, number of variables, box constraints used and references are collected in Table 4. If only one interval is given for the box constraints in Table 4, then each variable was restricted to the same interval. The test problems are the same as used in [24]. They are divided into two problem sets according to the number of variables so that problems with 10
![Page 408: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/408.jpg)
or less variables are here called small and others large.
![Page 409: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/409.jpg)
Originally, initial populationswere generated in altogether 21 differentways includ-
![Page 410: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/410.jpg)
ing 13 variants of the SSI process, 5 quasi random number generators, a nonaligned systematic sampling and a pseudo random number generator. The variants of the SSI process differed in the random number generator used when generating the trial points and in the metric used when computing the probabilistic maximal minimum distance PMMD. The quasi random number generators tested were the Niederreiter,
![Page 411: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/411.jpg)
Tab le 4 Summary of the test problems
![Page 412: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/412.jpg)
Dimensions Box constraints
![Page 413: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/413.jpg)
Michalewicz
![Page 414: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/414.jpg)
6, 10, 20, 30 [−600, 400] 6, 10, 20, 50 [−500, 500]
![Page 415: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/415.jpg)
Branin rcos [−5, 10]×[0, 15] Griewangk 2, 6, 10, 20, 50 [−700, 500] Ackley’s Path 2, 6, 10, 20, 30 [−30.768, 38.768]
![Page 416: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/416.jpg)
[−100, 100]
![Page 417: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/417.jpg)
[−900, 1100]
![Page 418: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/418.jpg)
Langerman
![Page 419: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/419.jpg)
Funtion 10
![Page 420: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/420.jpg)
Epistatic Michalewicz
![Page 421: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/421.jpg)
J Glob Optim (2007) 37:405–436 423
4).
f (x)
problems
Pseudo Nieder SSI Nonalig
f (x)Small −186.7 −186.7 −186.8 −186.6 f (x)Large −1896.0 −2205.4 −1680.2 −2295.9
0.34 0.28 0.37 0.58 885.9 355.3 162.5 1697.9
![Page 422: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/422.jpg)
Faure and Halton generator and two variants of the Sobol’ generator. The numerical
![Page 423: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/423.jpg)
results of the genetic algorithm with quasi random initial populations are reported in [23], where the Niederreiter generator performed the best. Here, we report the results of the genetic algorithm using initial populations generated by the good rep-
![Page 424: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/424.jpg)
resentatives from different types of point generators (see the beginning of Section
![Page 425: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/425.jpg)
5.2 Numerical results
![Page 426: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/426.jpg)
In this subsection, we present the results of the experiments described above. In what
![Page 427: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/427.jpg)
follows, we consider only the best objective function values that the algorithm has
![Page 428: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/428.jpg)
found (in the last population) and pay attention to the average and variance of these
![Page 429: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/429.jpg)
values in the repeated runs.
![Page 430: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/430.jpg)
Let us first consider those tests where the geneticalgorithm was run until a stopping
![Page 431: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/431.jpg)
criterion was satisfied. Table 5 shows the average final objective function values
![Page 432: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/432.jpg)
and the average standard deviations σ(f )for small and largeproblems.As we can see,
![Page 433: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/433.jpg)
for smallproblems, in the average objective function values, there are no differences in three significant digits. However, for large problems the differences in average objec- tive function values are quite large. Based solely on the average objective function
![Page 434: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/434.jpg)
values in Table 5 one could draw the conclusion that nonaligned systematic samples
![Page 435: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/435.jpg)
and Niederreiter quasi random points are clearly the most suitable for initial popula-
![Page 436: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/436.jpg)
tions. This, however, is not completely true as our further analysis reveals, when we
![Page 437: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/437.jpg)
consider also the variances.
![Page 438: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/438.jpg)
Before any statistical tests, we make an observation on the results in Table 5. As
![Page 439: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/439.jpg)
mentioned in Section 4.2, nonaligned systematic sampling reduces to pseudo random sampling for large problems. However, the average objective function value and the
![Page 440: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/440.jpg)
average standard deviation of those algorithms differ quiteremarkably for large prob-
![Page 441: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/441.jpg)
lems. This indicates that the average values are not good measures in the comparison.
![Page 442: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/442.jpg)
We applied analysis of variance (ANOVA) to the final objective function values to
![Page 443: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/443.jpg)
find out whether the differences were statistically significant. The analysis of variance
![Page 444: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/444.jpg)
indicated that only in 4 out of 52 test problems there were statistically significant
![Page 445: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/445.jpg)
differences (with95% confidence).We call these four problems critical problems,and
![Page 446: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/446.jpg)
they are the problems 40, 49, 51 and 52 in Table 4. Surprisingly, all the critical prob-
![Page 447: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/447.jpg)
lems were small (problems with 10 or less variables). This can be explained by the small variance for these problems. The fact that there was no statistically significant
![Page 448: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/448.jpg)
difference in the objective function values for any of the large problems is explained by their very large variances. The variances were the largest for the 20-dimensional
![Page 449: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/449.jpg)
Katsuura problem for which the variances were 8,473, 2,060, 509 and 17,100 for the pseudo, Niederreiter, SSI and nonaligned, respectively. The 20-dimensional Katsuura
![Page 450: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/450.jpg)
Tab le 5 Average objective
![Page 451: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/451.jpg)
function values and standard
![Page 452: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/452.jpg)
deviations for small and large
![Page 453: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/453.jpg)
σ(f )Small
![Page 454: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/454.jpg)
σ(f )Large
![Page 455: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/455.jpg)
J Glob Optim (2007) 37:405–436 424
in one.
sf =f − f min
f max − f min
,
-0.8
-0.6
Ps Ni SSI No
-14
-10
-6
-2
Ps Ni SSI No -4.7
-4.6
-4.5
Ps Ni SSI No
-8.8
Ps Ni SSI No -1
Problem 51 (P=0.001)
-9.6
-9.2
![Page 456: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/456.jpg)
problem biased the average values in Table 5 and was the main cause of differences
![Page 457: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/457.jpg)
for large problems.
![Page 458: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/458.jpg)
Next, we study the critical problems more closely. Figure 9 shows the average
![Page 459: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/459.jpg)
objective function values for the four critical problems. Ps, Ni, SSI and No stand for
![Page 460: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/460.jpg)
the pseudo, Niederreiter, SSI process and nonaligned systematic sampling, respec-
![Page 461: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/461.jpg)
tively, and the whiskers illustrate the standard deviations. In addition, the P values related to the F test of ANOVA [28] are given in the parenthesis subsequent to the problem number. The four plots reveal that there is no overall dominance between
![Page 462: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/462.jpg)
the geneticalgorithm variants, however the variant applying quasirandom points per-
![Page 463: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/463.jpg)
formed best in 3 out of 4 cases and also points generated by the SSI process performed better than pseudo random points in 3 out of 4 instances. Nonaligned systematic sam-
![Page 464: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/464.jpg)
pling performed very similarly to pseudo random points in three cases and was better
![Page 465: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/465.jpg)
So far we have studied the differences in objective function values after running
![Page 466: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/466.jpg)
the whole algorithm and there has been only minor differences. Next, we consider also the situations, where the algorithm is stopped prematurely after 10 and 20 gener-
![Page 467: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/467.jpg)
ations. Earlier, we noticed difficulties in interpreting values that were not scaled: one
![Page 468: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/468.jpg)
large value could strongly bias the results. Therefore, we now define a more reliable measure of performance by scaling the objective function values to the range from 0
![Page 469: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/469.jpg)
to 1. Scaled average objective function value sf is defined as follows
![Page 470: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/470.jpg)
Problem 40 (P=0.007)
Problem 49 (P=0.040)
![Page 471: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/471.jpg)
Problem 52 (P=0.043)
![Page 472: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/472.jpg)
Fig. 9 Average objective function values and standard deviations
![Page 473: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/473.jpg)
J Glob Optim (2007) 37:405–436 425
min
max
max
min
min
max
f max − f min
results.
0.90 0.91 0.85 0.88 0.9 0.87
all 0.52 0.38 0.84 0.85 0.91 0.90
all 0.45 0.61
![Page 474: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/474.jpg)
where f is the best objective function value known (or found after running all the
![Page 475: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/475.jpg)
four whole algorithms), and f is the worst of the best objective function value
![Page 476: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/476.jpg)
found during the repeated runs (either for a fixed number of generations or till a
![Page 477: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/477.jpg)
stopping criterion). It is worth noting that f changes each time the stopping criteria are changed, but f typically remains the same (at least if it is known). Finally, a
![Page 478: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/478.jpg)
bar above a function symbol denotes the average value over all the considered test problems (small or large), where the difference between (unscaled values of) f and f was more than 0.1. Table 6 shows the scaled average objective function values for
![Page 479: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/479.jpg)
small and large problems.
![Page 480: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/480.jpg)
We remind again that for large dimensional problems the nonaligned systematic
![Page 481: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/481.jpg)
sampling points and pseudo random points are essentially the same, and their results
![Page 482: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/482.jpg)
are here reported only for the sake of completeness. We now notice in Table 6 that
![Page 483: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/483.jpg)
pseudo and nonaligned systematic sampling receive similar values, which gives indi-
![Page 484: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/484.jpg)
cation that the values are more reliable after scaling.
![Page 485: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/485.jpg)
The scaled average objective function value sf becomes unstable near theoptimum
![Page 486: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/486.jpg)
since the divisor approaches zero and, therefore, the values for 10 and
![Page 487: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/487.jpg)
20 generations are more reliable than the values after running the whole algorithm. Nevertheless, Table 6 shows an interesting trend: the genetic algorithm variants that converge relatively fast in the first generations are not necessarily the ones obtain-
![Page 488: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/488.jpg)
ing the best final results. This is particularly true for the SSI process, which had the
![Page 489: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/489.jpg)
best empty space statistic values in Section 4 and in the numerical results in Table 6 receives worst values during the first generations, but good values at the end. Pseudo random points, on the other hand, had the best genetic diversity, and, in Table 6, they receive good values during the first iterations. The Niederreiter quasi random points had above average genetic diversity and above average coverage and their performance in numerical tests in Table 6 is above average except for one instance.
![Page 490: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/490.jpg)
Despite the observations made above, the differences in the results shown in Table 6
![Page 491: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/491.jpg)
are debatable.
![Page 492: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/492.jpg)
We applied ANOVA also for the intermediate results after 10 and 20 generations.
![Page 493: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/493.jpg)
Again, we call those problems critical, where there were statistical differences in the objective function values. However, we now exclude the problems, where the differ- ences in objective function values were less than 0.1. There were all together 38 critical problems after 10 generations and 9 after 20 generations. In Table 7, we report the number of critical problems for which each algorithm had the best average interme- diate objective function value. The differences are not large, but pseudo random (and
![Page 494: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/494.jpg)
nonaligned systematic sampling) initial population seem to provide best intermediate
![Page 495: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/495.jpg)
Tab le 6 Scaled average objective function values after
Generations Pseudo Nieder SSI Nonalig 10 and 20 generations and after running the whole algorithm sf (x)Small 10
sf (x)Small 20 0.95 0.89
![Page 496: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/496.jpg)
sf (x)Small 0.43 0.57 sf (x)Large 10 0.99 0.84 sf (x)Large 20 0.99 0.87 sf (x)Large 0.38 0.42
![Page 497: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/497.jpg)
J Glob Optim (2007) 37:405–436 426
Total
10 9 9 8 12 38 20 5 1 0 3 9
Total
10 18 11 8 9 46 20 12 11 6 15 44 Final 5 9 3 2 19
![Page 498: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/498.jpg)
Tab le 7 The number of problems with best average
Generations Pseudo Nieder SSI Nonalig
![Page 499: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/499.jpg)
objective function values for
![Page 500: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/500.jpg)
critical problems
![Page 501: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/501.jpg)
Before summarizing the results, let us yet consider the magnitudes of the variances
![Page 502: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/502.jpg)
in the best objective function values separately from the actual objective function
![Page 503: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/503.jpg)
values. In Table 8 are reported the number of test problems for which each algorithm had the smallest variance after 10 and 20 generations and after running the whole algorithms. We have excluded the problems where the differences in variances were less than 0.1. The total number of problems considered is reported in the last column of Table 8. It is noteworthy that the variant of genetic algorithm using pseudo random numbers has most often the smallest variance for the intermediate solutions, which
![Page 504: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/504.jpg)
was not expected.
![Page 505: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/505.jpg)
5.3 Summary of numerical results
![Page 506: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/506.jpg)
Here, we summarize the analysis made above.When considering final and intermedi-
![Page 507: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/507.jpg)
ate average objective function values, the analysis of variance indicated four critical
![Page 508: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/508.jpg)
problems for the final results and many more for the intermediate results where there
![Page 509: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/509.jpg)
were significant differences in the best objective function values found. None of the methods clearlydominated the others, but in the scaled objective function values there
![Page 510: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/510.jpg)
was a noticeable trend, namely that pseudo random initial populations provided on the average good intermediate results, but not so good final results. The converse was true for the SSI process and quasi random points with better uniform coverage. Ear-
![Page 511: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/511.jpg)
lier we noticed that some clusterizations were harmful and some beneficial. Hence,
![Page 512: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/512.jpg)
genetic algorithms using pseudo random numbers and the variants where the initial
![Page 513: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/513.jpg)
populations have better uniform coverage performed in a slightly different way. Nev- ertheless, the differences, in general, were debatable and no strong conclusions could
![Page 514: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/514.jpg)
be made.
![Page 515: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/515.jpg)
As far as the magnitudes of variances in the intermediate results are concerned,
![Page 516: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/516.jpg)
in most of the cases, the variance was the smallest for the genetic algorithm using
![Page 517: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/517.jpg)
pseudo random numbers. There may be both harmful and beneficial clusterizations
![Page 518: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/518.jpg)
as seen in Figs. 1 and 2. Harmful clusterizations lead to inferior and beneficial lead to superior objective function values compared to initial populations without clustering. One could have expected this to imply more significant differences in the magnitudes
![Page 519: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/519.jpg)
of variances in the early generations but this did not happen. However, again, none
![Page 520: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/520.jpg)
of the algorithms clearly dominated the others and the differences were not large.
![Page 521: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/521.jpg)
Finally, there were no significant differences in the magnitudes of variances related
![Page 522: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/522.jpg)
to final objective function values when applying clustered or unclustered initial popu-
![Page 523: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/523.jpg)
Tab le 8 Number of test problems when the variance
Generations Pseudo Nieder SSI Nonalig
![Page 524: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/524.jpg)
was smallest
![Page 525: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/525.jpg)
J Glob Optim (2007) 37:405–436 427
b) a)
![Page 526: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/526.jpg)
lations.This could be assumed keeping inmind the observation that geneticalgorithms
![Page 527: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/527.jpg)
areMarkov chains and, hence, converge independently of the initial population.How-
![Page 528: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/528.jpg)
ever, the conclusion is not trivial, since actually theMarkov chain property only guar-
![Page 529: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/529.jpg)
antees convergence when thenumber of generations approaches infinity. The last row
![Page 530: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/530.jpg)
in Table 8 shows that the geneticalgorithm applying pseudorandom initial population
![Page 531: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/531.jpg)
has more often smallest variance than the one applying the SSI process, but less often than the one applying quasi random sequences. Hence, the clusterization of the initial population did not seem to have strong effect on the variance of the final objective
![Page 532: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/532.jpg)
function values.
![Page 533: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/533.jpg)
Despite some differences, the results indicate that pseudo random points perform
![Page 534: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/534.jpg)
rather well when used in initial populations of genetic algorithms. Moreover, if the
![Page 535: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/535.jpg)
minor differences found in this research can be generalized, then they speak for
![Page 536: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/536.jpg)
pseudo random points in cases when the algorithm is stopped after relatively small
![Page 537: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/537.jpg)
number of generations, which may be the case with some computationally expensive
![Page 538: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/538.jpg)
real life problems.
![Page 539: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/539.jpg)
6 Discussion
![Page 540: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/540.jpg)
In this section, we consider fourproblematic issues that we have touched earlier. First, we point out that dimensionality plays a large role in this research. The initial popula- tions become sparse when the optimization problem has more than few variables. A
![Page 541: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/541.jpg)
simple example of this can be seen with unit hyper cubes in three and two dimensions.
![Page 542: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/542.jpg)
In three dimensions, eight point can be located in the corners of the cube so that
![Page 543: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/543.jpg)
their maximal minimum distance is always one. But locating eight points in the line
![Page 544: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/544.jpg)
segment of length one means that the maximal minimum distance between the points is 1/7. The sparsity in higher dimensions means that solutions in the initial population are likely to be relatively far from the global optimum no matter what uniform distri- bution pattern is used. Moreover, in sparse populations, when the minimum distance
![Page 545: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/545.jpg)
between two adjacent points in the population is maximized, that is, when the points
![Page 546: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/546.jpg)
in the population try tomaximally avoid each other, then the population is likely to be
![Page 547: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/547.jpg)
concentrated to the borders of the feasible region, whichmay not always be desirable.
![Page 548: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/548.jpg)
This phenomenon is typical of the SSI process and could also be seen in the example
![Page 549: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/549.jpg)
mentioned above.
![Page 550: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/550.jpg)
The second issue is of more technical nature and concerns the SSI process. The SSI
![Page 551: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/551.jpg)
process can produce points that are either genetically diverse or have good uniform coverage, depending on the distance parameter PMMD. We optimized the coverage, but did not test values forPMMD that would compromise between the two conflicting objectives. Smaller values for would also make the process considerably faster. For example, in Fig. 10a we have seven point in an interval of length one with = 1/7.
![Page 552: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/552.jpg)
We know that one more point can be included but how many random points should
![Page 553: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/553.jpg)
be generated before the right location is found?
![Page 554: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/554.jpg)
When optimizing the coverage for the SSI process the automatic determination of
![Page 555: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/555.jpg)
a maximal minimum distance PMMD is problematic. We did not find good theoret-
![Page 556: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/556.jpg)
ical estimates for PMMD and the theoretical estimates we tested did not work well
![Page 557: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/557.jpg)
Fig. 10 Difficulties with speed
![Page 558: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/558.jpg)
and approximation of
![Page 559: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/559.jpg)
J Glob Optim (2007) 37:405–436 428
![Page 560: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/560.jpg)
in practice. Hence, we used experimental estimates. In our current implementation,
![Page 561: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/561.jpg)
we have numerically solved maximal values of PMMD for some number of points
![Page 562: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/562.jpg)
in some dimensions, and we approximate maximal PMMD elsewhere using interpo-
![Page 563: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/563.jpg)
lation. Examples of values used for in our experiments for different numbers of variables (with population size 200) include 0.032 (for n = 2), 0.23 (for n = 5), 0.49 (for n = 10), 0.64 (for n = 20), 0.79 (for n = 50) and 0.84 (for n = 100). Some further research is needed before the SSI process with maximal values for PMMD can be
![Page 564: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/564.jpg)
used in practical applications.Anexample of the difficulty of approximating the value
![Page 565: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/565.jpg)
of is given in Fig. 10b. Let us assume that we have fixed = 1 / 7 a nd t h e d istance
![Page 566: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/566.jpg)
between the end points of the interval and outermost points is 1 / 8 a nd t h e d istance
![Page 567: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/567.jpg)
between intermediate points is 2/8. Here we have only four points (denoted by black
![Page 568: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/568.jpg)
dots in the figure) and cannot fit any more points even though eight points could be
![Page 569: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/569.jpg)
fit in the interval in the optimal case (denoted by circles in the figure). In other words, it is very hard to determine the maximal minimum distance a priori when points are
![Page 570: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/570.jpg)
generated randomly.
![Page 571: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/571.jpg)
The third issue concerns the speed of point generators.We point out that the com-
![Page 572: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/572.jpg)
putational complexity of an algorithm may depend strongly on the implementation,
![Page 573: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/573.jpg)
see, e.g., [4], where bit-operations are considered. Thus, the results on the speed of
![Page 574: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/574.jpg)
the generators can be considered only suggestive.
![Page 575: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/575.jpg)
The last issue concerns the testing of point generators. There exists a large variety
![Page 576: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/576.jpg)
of dynamic statistical tests for the independence of points generated by point gener- ators [14]. These tests are called goodness-of-fit tests. One well-established battery of goodness-to-fit tests is called DIEHARD and the source code is available at [8]. However, the goodness-to-fit tests are designed to test the independence of large
![Page 577: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/577.jpg)
numbers of points. For example, DIEHARD requires a binary file of a size 10–11 megabytes to evaluate a generator. Since we are only interested in a few hundred
![Page 578: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/578.jpg)
numbers at the beginning of the sequence, we used the Ripley’s K-function instead of
![Page 579: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/579.jpg)
the goodness-to-fit tests.
![Page 580: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/580.jpg)
7 Conclusions and future research
![Page 581: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/581.jpg)
In this paper, we have tested different initial populations for a real coded genetic
![Page 582: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/582.jpg)
algorithm. Traditionally, pseudo random numbers are used for generating initial pop-
![Page 583: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/583.jpg)
ulation, and our motivation has been to study and initiate discussion on whether their
![Page 584: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/584.jpg)
use is justified.
![Page 585: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/585.jpg)
We have shown with asimple, academic,example that initial populationsmay have
![Page 586: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/586.jpg)
a significant effect on the best objective function value over several generations. Then
![Page 587: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/587.jpg)
we have concentrated on studying different realistic ways to generate initial popu-
![Page 588: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/588.jpg)
lations for a case with no a priori information on the location of the global minima. We have briefly summarized the basic properties of a pseudo and a quasi random sequence generator, the SSI process and the nonaligned systematic sampling and applied them. We have discussed their properties including genetic diversity and a gooduniform coverage for the points as well as speed and usability for the generators. In thenumerical tests with geneticalgorithms, we have used a test suite of 52 functions
![Page 589: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/589.jpg)
from the literature.We have studied the effects of the different initial populations on
![Page 590: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/590.jpg)
the best objective function values after 10 and 20 generations and after running the
![Page 591: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/591.jpg)
whole algorithm.
![Page 592: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/592.jpg)
J Glob Optim (2007) 37:405–436 429
102020.
![Page 593: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/593.jpg)
There were differences in the coverage and genetic diversity of the tested point
![Page 594: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/594.jpg)
sets. The SSI process has a good uniform coverage, but only average genetic diversity, and for pseudo random points it is vice versa. The Niederreiter quasi random points
![Page 595: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/595.jpg)
have both above average coverage and above average genetic diversity.
![Page 596: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/596.jpg)
In the numerical experiments with the genetic algorithms, there was a trend—
![Page 597: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/597.jpg)
although weak—showing that the versions of genetic algorithm with good genetic diversity converged fast during the first generations, but did not obtain the best final objective function values on the average. The converse was true for the versions with good uniform coverage. However, the differences were not so large that any strong
![Page 598: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/598.jpg)
conclusion could be drawn.
![Page 599: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/599.jpg)
With respect to the speed and usability of the point generators the results show that
![Page 600: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/600.jpg)
pseudo and quasi random sequence generators are fast and easy to use. Both the SSI process and the nonaligned systematic sampling require more developing and testing.
![Page 601: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/601.jpg)
We conclude that, based on our research, the traditional way to generate initial
![Page 602: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/602.jpg)
populations of genetic algorithms using pseudo random number generator was not worse than the others. It is particularly well suited to cases where the algorithm must be stopped prematurely; which may happen with computationally expensive real
![Page 603: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/603.jpg)
life problems. However, there are also good alternative ways such as quasi random
![Page 604: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/604.jpg)
sequences and the SSI process, which have advantages in certain cases, especially if the goodness of the final solution is valued higher that the speed of convergence
![Page 605: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/605.jpg)
during the first generations.
![Page 606: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/606.jpg)
Our findings show that paying attention to the initial population may have an
![Page 607: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/607.jpg)
effect on the success of the genetic algorithm and further research and discussion is encouraged. The topics for future research include, firstly, to study different ini- tial populations for specific types of problems and with different genetic algorithm
![Page 608: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/608.jpg)
parameters like population size and maximum number of generations. Secondly, to
![Page 609: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/609.jpg)
study more closely the problem of dimensionality discussed in Section 6, and thirdly, to further develop the SSI implementation and to discover good theoretical estimates
![Page 610: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/610.jpg)
for the maximal minimum distance PMMD.
![Page 611: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/611.jpg)
Acknowledgements The authors thank doctors Salme Kärkkäinen andMarkoM.Mäkelä for useful discussions and professor Michael Mascagni for providing the Niederreiter generator. This research was supported by the Jenny and AnttiWihuri Foundation and the Academy of Finland, grant number
![Page 612: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/612.jpg)
Appendix A
![Page 613: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/613.jpg)
This appendix is designed to give a short overview to different pseudorandomnumber and quasi random sequence generators. In the following definitions we use (y)modM
![Page 614: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/614.jpg)
to denote y modulo M.
![Page 615: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/615.jpg)
A.1 Some pseudo random number generators
![Page 616: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/616.jpg)
Traditionally, pseudo random number generators produce sequences of scalars. If vectors are needed, then they are usually formed by taking the first n scalars for the first n-dimensional vector, the next n scalars for the second vector and so forth. According to [30], it would be preferable to generate pseudo random vectors directly. Here, we present traditional pseudo random number generators and one vector
![Page 617: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/617.jpg)
generator.
![Page 618: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/618.jpg)
i
0 = y0 i
)
congruential generators with a = 0 a nd a =0. Then a generator with a =0 is called a
i = (a i−1
2 2 2
1
J Glob Optim (2007) 37:405–436
2 ˆi ˆy = (a ˆi
430
j
0
i
i ˆ i
Linear y 1 y 2 )modM
y i = (a 1 (y i−1 2 y +a 3 )modM
y i = (a 1 y 1 i−1 2
y i−1
y i +y i−1
x = k i y i
M
1
)modM 2
3 ̂ y −1 )modM 3
x y i i = ( M 1
ˆyi + M 2
ˆi ˆy + M 3 )mod 1.
2
j i
![Page 619: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/619.jpg)
We classify pseudo random number generators to two main categories: congruen-
![Page 620: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/620.jpg)
tial and recursive generators.We call amethod congruential, if it usesmodulo and only the previous iteration value, andwe call it recursive, if it uses values from several previ-
![Page 621: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/621.jpg)
ous iterations. In addition, we present a feedback shift register generator and a vector generator called SQRT-generator, which do not fall into the two main categories. For another classification, see, e.g., [39]. The classifications are always somewhat vague.
![Page 622: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/622.jpg)
Feedback shift register generator also uses values from several previous iterations,
![Page 623: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/623.jpg)
but its construction differs significantly from recursive generators, and therefore, we
![Page 624: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/624.jpg)
put it in its own class. For more detailed information about pseudo random number
![Page 625: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/625.jpg)
generation we refer to [14] and to references therein.
![Page 626: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/626.jpg)
A.1.1 Congruential generators
![Page 627: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/627.jpg)
Next, we present five congruential generators. They are called linear, quadratic, inver-
![Page 628: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/628.jpg)
sive, additive and parallel linear congruential generators. The name congruential
![Page 629: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/629.jpg)
comes from the use of modulo. For example, in modulo 5, the numbers 4 and 9 are called congruent. In what follows,M is themodulo and a ’s are prescribed integers.On
![Page 630: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/630.jpg)
the ith iteration, the generators first produce an integer y i ∈ [ 0,M)and then arandom number x i ∈ [ 0, 1)is obtained by a division xi = yi /M unless another formula is given.
![Page 631: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/631.jpg)
The seed y is a large prescribed integer. An additive congruential generator (of kth
![Page 632: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/632.jpg)
order) [45], to be described below, requires k 0 +1 seeds 0 ≤ yj < M, j = 0, ... , k ,a nd
![Page 633: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/633.jpg)
the parallel linear generator includes three separate linear generators denoted by y ,
![Page 634: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/634.jpg)
ˆy and ˆy .
![Page 635: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/635.jpg)
i = (a i−1 +a
![Page 636: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/636.jpg)
Quadratic
2 +a i−1
Inversive
+a )modM
![Page 637: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/637.jpg)
Additive
![Page 638: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/638.jpg)
m = (ym−1 m )modM,m = 1, ..., k
![Page 639: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/639.jpg)
Parallel linear y
y )modM
![Page 640: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/640.jpg)
ˆyi = (a ˆyi−1
![Page 641: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/641.jpg)
In all of the above generators i = 1, 2, ... Often, a distinction is made between linear
multiplicative linear congruential generator and a generator with a =0 a mixed linear
![Page 642: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/642.jpg)
congruential generator.
![Page 643: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/643.jpg)
A.1.2 Recursive generators
![Page 644: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/644.jpg)
Generators using two or more previous random numbers when generating a new number in the sequence are here called recursive generators. In what follows, a , c , s,
![Page 645: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/645.jpg)
J Glob Optim (2007) 37:405–436
−y i− r −c i
of zeros and ones.
p ∈ [ 0,M) can be selected arbitrarily.
431
i
i = yi
y 1 y k y i−k )modM
y )modM
y +y i− r +c i
c =i
i = (yi−s
1, y
c =i 1, if y i−s < 0
y 1 y +a i−r 2 y +c i i is i−1 i
Again i
j
i j
p 1
p−1 +· · · +a −1z +a ) , ( 1) p p
i i
n p
n−p p
n−p+1 +· · · +a α ) m o d M . ( 2) 1 n−1
1
![Page 646: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/646.jpg)
and r are prescribed integers s < r, j
= 1, ... , k , a nd ci is called the carry operator on
![Page 647: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/647.jpg)
the ith iteration. The determination of c is omitted for the multiply-with-carry-gener- ator and also for the add-with-carry and substract-with-borrow generators usingmore
![Page 648: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/648.jpg)
that two previous values. Again, the pseudo random number x i ∈ [ 0, 1)is obtained by a division x /M. For more details about the carry operator, see, e.g., [6]. Next, we
![Page 649: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/649.jpg)
present five recursive generators.
![Page 650: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/650.jpg)
Multiplicative recursive
i = (a i−1 +· · · +a
Lagged Fibonacci
i = (yi−r −y i− s
Add −with −carry ) m o d M , w here
![Page 651: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/651.jpg)
i−s +y i− r +c i−1 ≤ M
![Page 652: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/652.jpg)
0, otherwise
![Page 653: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/653.jpg)
Substract
−with −borrow yi = (yi−s −y i− r −c i ) m o d M , w here
![Page 654: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/654.jpg)
0, otherwise
![Page 655: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/655.jpg)
Multiply −with −carry i = (a i−s )modM ,w here c
![Page 656: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/656.jpg)
computed using c and previous values of y .
![Page 657: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/657.jpg)
= 1, 2, ... and y0
is the seed. The number of previous values of y used in the
![Page 658: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/658.jpg)
methods described above may alter. In addition, the lagged Fibonacci generator may
![Page 659: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/659.jpg)
generate new values also as a sum or a product of the previous values.
![Page 660: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/660.jpg)
A.1.3 Feedback shift register generator
![Page 661: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/661.jpg)
Feedback shift register generator uses modulo in a different manner compared to the congruential and recursive generators. Modulo M is not a large integer, but very
![Page 662: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/662.jpg)
often M = 2 in which case the generator produces a sequence α
![Page 663: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/663.jpg)
The sequence is then cut into subsequences with an appropriate length. These subse- quences correspond to y ’s. To generate the sequence α , first a primitivepolynomial
![Page 664: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/664.jpg)
q (z ) of order p is selected
![Page 665: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/665.jpg)
q(z) = z −(a z
![Page 666: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/666.jpg)
where a ∈{0, 1, ... ,M −1}, i = 1, ... , p. Then using the coefficients a , t he nth num-
![Page 667: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/667.jpg)
ber in the sequence is generated using the formula
![Page 668: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/668.jpg)
α = (a α
+a − 1α
![Page 669: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/669.jpg)
The initial numbers α , ... , α
![Page 670: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/670.jpg)
J Glob Optim (2007) 37:405–436
i √p , i = 1, ... , n. Then the n-dimensional SQRT sequence
432
i i
i 1 n
E =n
i=1
a i
b d i ,
a +1 i
b d i ,
i i i d i i
n ⊂ Rn d i
1 2 1 2
t
t−m 2
m
τ
1
1
![Page 671: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/671.jpg)
A.1.4 SQRT sequence
![Page 672: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/672.jpg)
The following definition of the SQRT sequence is taken from [43]. Let p now be the
![Page 673: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/673.jpg)
ith prime number and z =
![Page 674: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/674.jpg)
{x i} is defined by
![Page 675: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/675.jpg)
x = (iz) = (iz , ... , iz )mod 1,
![Page 676: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/676.jpg)
where themodulo is takencomponent-wise.Despite itssimplicity, the SQRT sequence
![Page 677: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/677.jpg)
performs well in [43], where it is applied to a numerical integration problem and com-
![Page 678: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/678.jpg)
pared with five quasi random sequences.
![Page 679: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/679.jpg)
A.2 Some quasi random sequence generators
![Page 680: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/680.jpg)
To describe the structure and the distribution properties of some quasi random se-
![Page 681: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/681.jpg)
quences let us define an elementary interval in base b
![Page 682: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/682.jpg)
where a , d and b are integers d ≥ 0, 0 ≤ a < b
for i = 1, ... , n. Thus, E is a subinterval of an n-dimensional unit cube I and its ith side has a length of
![Page 683: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/683.jpg)
1/b . Figure 11 illustrates 2-dimensional elementary intervals with base b = 2 a nd
![Page 684: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/684.jpg)
d = d = 2, and the colored area corresponds to values a = 1 a nd a = 2.
![Page 685: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/685.jpg)
Quasi random sequences are called (t, s)-sequences if they satisfy the following
![Page 686: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/686.jpg)
distribution property (s is the dimension that we denote by n). For all integers k ≥ 0,
![Page 687: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/687.jpg)
the point set {xj} of (t, s)-sequence with kbm ≤ j < ( k + 1)bm has exactly b points
![Page 688: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/688.jpg)
on every elementary interval in base b with volume b (in L -metric) [36]. The distribution properties for a (t, s)-sequence are the most preferable, when the dis-
![Page 689: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/689.jpg)
tribution parameter t = 0, because then the first bm successive points divide evenly
![Page 690: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/690.jpg)
on the corresponding elementary intervals and the same is true for the following b
![Page 691: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/691.jpg)
successive points and so on.
![Page 692: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/692.jpg)
The class of (t, s)-sequences was introduced by Sobol’ in 1966 (see, [36]). He called
![Page 693: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/693.jpg)
them LP -sequences (τ ≡ t) and studied them in base 2. Faure generalized the
![Page 694: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/694.jpg)
Fig. 11 Two-dimensional
![Page 695: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/695.jpg)
elementary intervals
![Page 696: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/696.jpg)
then Van der Corput sequence in base b is the sequence {φ (k)} . Informally, we ∞k=0
may say that in the radical inverse the numbers d , ... , d are reflected using the deci- 1 j
J Glob Optim (2007) 37:405–436
2 2 2
der Corput sequence is a 1-dimensional (t, s)-sequence with t = 0 (see, e.g., [30]).
433
j 1 b
i b
φ (k ) =b d 1
b
d j
b j = (0.d ... d ) , ( 3) 1 j b
b
i b 1 b −1 n
(i)), i = 0, 1, ... ,N − 1 ( 4)
1 n
i b 1 b n (i)), i = 0, 1, ... . ( 5)
i i
p 1
p−1 +· · · +a − 1 z + a . ( 6) p p
![Page 697: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/697.jpg)
(t, s)-sequences to arbitrary prime base b ≥ 2 (see, e.g., [36]). Finally, Niederreiter
![Page 698: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/698.jpg)
generalized (t, s)-sequence to arbitrary base b ≥ 2.
![Page 699: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/699.jpg)
Often a (t, s)-sequence is called the Sobol’ sequence if b = 2 a nd t = 0, and Faure
![Page 700: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/700.jpg)
sequence if b is a prime and t = 0 [30]. Sequences that are constructed according to the
![Page 701: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/701.jpg)
guidelines given in [30] are called Niederreiter sequences. For a general construction
![Page 702: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/702.jpg)
of any (t, s)-sequence, see, e.g., [20, 30].
![Page 703: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/703.jpg)
Next, we present some examples how to construct (t, s)-sequences. We omit all
![Page 704: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/704.jpg)
implementational details.
![Page 705: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/705.jpg)
A.2.1 Van der Corput sequence
![Page 706: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/706.jpg)
Let b ≥ 2 be the base and k be an integer. If we write k in base b
k = (d ...d ) ,
where d ∈ { 0, ... , b −1}, i = 1, ... , j, and define a radical inverse function φ by
![Page 707: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/707.jpg)
+· · · +
![Page 708: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/708.jpg)
mal point as a reflector, for example, if k = 12 = (1100) , t hen φ (12) = 0.0011 .Van
![Page 709: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/709.jpg)
A.2.2 Hammersley sequence
![Page 710: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/710.jpg)
All but the first component of the n-dimensional Hammersley sequence are van der Corput sequences in an appropriate base. The n-dimensional Hammersley sequence
![Page 711: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/711.jpg)
is defined as
![Page 712: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/712.jpg)
x = (i/N, φ (i), ... , φ
![Page 713: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/713.jpg)
where N is the number of points to be generated and the bases b , ... , b are integers
![Page 714: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/714.jpg)
greater than one and pairwise prime.
![Page 715: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/715.jpg)
A.2.3 Halton sequence
![Page 716: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/716.jpg)
A Hammersley sequence without the first element is called Halton sequence [17]
![Page 717: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/717.jpg)
x = (φ (i), ... , φ
![Page 718: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/718.jpg)
The advantage over the Hammersley sequence is that a Halton sequence does not
![Page 719: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/719.jpg)
require the knowledge of N, the total length of the sequence.
![Page 720: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/720.jpg)
A.2.4 Sobol’ sequence
![Page 721: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/721.jpg)
We construct the Sobol’ sequence following [4] and [14]. Let us first consider 1-dimensional Sobol’ sequence, for which we need to create a set of direction num- bers v in base 2. To create the direction numbers v , we use coefficients of a primitive
![Page 722: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/722.jpg)
polynomial q in the field of binary numbers (compare to Eq. 1 in Sect. 7)
![Page 723: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/723.jpg)
q(z) = z +a z
![Page 724: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/724.jpg)
J Glob Optim (2007) 37:405–436
3 2 1 b = 1, b = 0, b = 1 a nd b = 1). To generate an n-dimensional sequence, it is
d mod b.
434
i 1
i−1 2
i−2 p
i−p v i−p
⊕2 p
, i > p.
i 1
1 2
2 3
3
4 3 2 1
b c 0
b c −1 j
b j ,
b 1 j b
c m
=
j
n=m
h!h
i b b
n−1 (φ (i))), b
(1994)
![Page 725: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/725.jpg)
Then, employing the bitwise binary exclusive-or operator ⊕ we derive the direction
![Page 726: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/726.jpg)
numbers from the formula
![Page 727: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/727.jpg)
v = a v ⊕a v ··⊕ · ⊕a v
![Page 728: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/728.jpg)
The ith recurrence in 1-dimensional Sobol’ sequence is now formed as
![Page 729: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/729.jpg)
x = b v ⊕b v ⊕b v ··⊕ · ,
![Page 730: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/730.jpg)
where ··· b b b is the binary representation of i (for example, if i = 1011, then
sufficient to choose n different primitive polynomials and calculate n different sets of
![Page 731: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/731.jpg)
direction numbers. For more details, see [4].
![Page 732: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/732.jpg)
A.2.5 Faure sequence
![Page 733: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/733.jpg)
The followingexample of Faure sequence ismainly from [43].Let us define a function g
![Page 734: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/734.jpg)
g(φ (k)) =+· · · +
![Page 735: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/735.jpg)
where b is the base, φ (k) = (0.d ... d ) is an element of the Van der Corput
![Page 736: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/736.jpg)
sequence and
![Page 737: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/737.jpg)
m(h −m) !
![Page 738: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/738.jpg)
Now, an n-dimensional (t, s)-sequence is defined as
![Page 739: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/739.jpg)
x = (φ (i), g(φ (i)), ... , g
![Page 740: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/740.jpg)
and a Faure sequence is obtained when b is the smallest prime larger or equal to the
![Page 741: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/741.jpg)
dimension n [36].
![Page 742: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/742.jpg)
References
![Page 743: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/743.jpg)
1. Acworth, P., Broadie, M., Glasserman, P.: A comparison of some Monte Carlo and quasi Monte
![Page 744: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/744.jpg)
Carlo techniques for option pricing. In: Niederreiter, H., Hellekalek, P., Larcher, G., Zinterhof,
![Page 745: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/745.jpg)
P. (eds). Monte Carlo and Quasi-Monte Carlo Methods 1996, number 127 in Lecture notes in
![Page 746: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/746.jpg)
statistics, pp. 1–18. Springer-Verlag, New York (1998)
![Page 747: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/747.jpg)
2. Ali, M.M., Storey, C.: Modified controlled random search algorithms. Int. J. Comput. Mathemat.
![Page 748: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/748.jpg)
53, 229–235 (1994)
![Page 749: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/749.jpg)
3. Ali,M.M., Storey, C.: Topographical multilevel single linkage. J. Global Optimizat. 5(4), 349–358
![Page 750: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/750.jpg)
4. Bratley, P., Fox, B.L.: Algorithm 659: implementing Sobol’s quasirandom sequence generator.
![Page 751: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/751.jpg)
ACM Trans. Mathemat. Software 14(1), 88–100 (1988)
![Page 752: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/752.jpg)
5. Bratley, P., Fox, B.L., Niederreiter, H.: Implementation and tests of low-discrepancy sequences.
![Page 753: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/753.jpg)
ACM Trans. Model. Comput. Simulat. 2(3), 195–213 (1992)
![Page 754: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/754.jpg)
6. Couture, R., L’Ecuyer, P.: Distribution properties of multiply-with-carry random number
![Page 755: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/755.jpg)
generators. Mathemat. Computat. 66(218), 591–607 (1997)
![Page 756: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/756.jpg)
7. Dekkers, A., Aarts, E.: Global optimization and simulated annealing.Mathemat. Program. 50(3),
![Page 757: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/757.jpg)
367–393 (1991)
![Page 758: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/758.jpg)
8. DIEHARD: a battery of tests for random number generators developed by George Marsaglia.
![Page 760: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/760.jpg)
9. Diggle, P.J.: Statistical Analysis of Spatial Point Patterns. Academic Press, London (1983)
![Page 761: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/761.jpg)
J Glob Optim (2007) 37:405–436 435
(2004)
(1998)
(2002)
(1989)
(1998)
![Page 762: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/762.jpg)
10. Dykes, S., Rosen, B.: Parallel very fast simulated reannealing by temperature block partitioning.
![Page 763: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/763.jpg)
In Proceedings of the 1994 IEEE International Conference on Systems, Man, and Cybernetics,
![Page 764: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/764.jpg)
vol. 2, pp. 1914–1919 (1994)
![Page 765: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/765.jpg)
11. Eiben,A.E.,Hinterding,R.,Michalewicz,Z.:Parameter control in evolutionaryalgorithms. IEEE
![Page 766: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/766.jpg)
Trans. Evol. Computat. 3(2), 124–141 (1999)
![Page 767: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/767.jpg)
12. Floudas, C.A., Pardalos, P.M. (eds.): RecentAdvances inGlobalOptimization. PrincetonUniver-
![Page 768: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/768.jpg)
sity Press, Princeton, NJ (1992)
![Page 769: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/769.jpg)
13. Genetic and evolutionary algorithm toolbox for use with Matlab. http://www.geatbx.com/, June
![Page 770: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/770.jpg)
14. Gentle, J.E.:RandomNumberGeneration andMonteCarloMethods. Springer-Verlag,NewYork
![Page 771: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/771.jpg)
15. Glover, F.,Kochenberger,G.A. (eds.):Handbook ofMetaheuristics.KluwerAcademic Publishers,
![Page 772: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/772.jpg)
Boston (2003)
![Page 773: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/773.jpg)
16. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-
![Page 774: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/774.jpg)
Wesley, New York (1989)
![Page 775: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/775.jpg)
17. Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-
![Page 776: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/776.jpg)
dimentional integrals. Numerische Mathematik, 2, 84–90 (1960)
![Page 777: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/777.jpg)
18. Horst,R., Pardalos, P.M. (eds.):Handbook ofGlobalOptimization.KluwerAcademic Publishers,
![Page 778: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/778.jpg)
Dordrecht (1995)
![Page 779: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/779.jpg)
19. Kingsley, M.C.S., Kanneworff, P., Carlsson, D.M.: Buffered random sampling: a sequential inhib-
![Page 780: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/780.jpg)
ited spatial point process applied to sampling in a trawl survey for northern shrimp pandalus
![Page 781: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/781.jpg)
borealis in west Greenland waters. ICES J. Mar. Sci. 61(1), 12–24 (2004)
![Page 782: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/782.jpg)
20. Larcher, G.: Digital point sets: analysis and applications. In: Hellekalek, P., Larcher, G. (eds.)
![Page 783: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/783.jpg)
Random and Quasi Random Point Sets, number 138 in Lecture Notes in Statistics, pp. 167–222.
![Page 784: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/784.jpg)
Springer-Verlag, New York (1998)
![Page 785: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/785.jpg)
21. Lei,G.:Adaptive random search inquasi-Monte Carlomethods for globaloptimization.Comput.
![Page 786: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/786.jpg)
Mathemat. Appl. 43(6–7), 747–754 (2002)
![Page 787: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/787.jpg)
22. Lemieux,C., L’Ecuyer, P.:On selection criteria for lattice rules and otherquasi-Monte Carlo point
![Page 788: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/788.jpg)
sets. Mathemat. Comput. Simulat. 55(1–3), 139–148 (2001)
![Page 789: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/789.jpg)
23. Maaranen, H.: On Global Optimization with Aspects toMethod Comparison and Hybridization.
![Page 790: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/790.jpg)
Licentiate thesis, Department of Mathematical Information Technology, University of Jyväskylä
![Page 791: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/791.jpg)
24. Maaranen, H., Miettinen, K., Mäkelä, M.M.: Quasi-random initial population for genetic algo-
![Page 792: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/792.jpg)
rithms. Comput. Mathemat. Appl. 47(12), 1885–1895 (2004)
![Page 793: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/793.jpg)
25. Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag,
![Page 794: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/794.jpg)
Berlin (1994)
![Page 795: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/795.jpg)
26. Michalewicz, Z., Logan, T.D., Swaminathan, S.: Evolutionary operators for continuous convex
![Page 796: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/796.jpg)
parameter spaces. In: Sebald, A.V., Fogel, L.J. (eds.) Proceedings of the 3rd Annual Conference
![Page 797: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/797.jpg)
on Evolutionary Programming, pp. 84–97.World Scientific Publishing, River Edge, NJ (1994)
![Page 798: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/798.jpg)
27. Morokoff, W.J., Caflisch, R.E.: Quasi-random sequences and their discrepancies. SIAM J. Sci.
![Page 799: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/799.jpg)
Comput. 15(6), 1251–1279 (1994)
![Page 800: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/800.jpg)
28. Neter, J.,Wasserman,W.,Whitmore,G.A.:Applied Statistics.Allyn andBacon, Inc.,Boston, third
![Page 801: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/801.jpg)
edition (1988)
![Page 802: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/802.jpg)
29. Niederreiter, H.: Quasi-Monte Carlo methods for global optimization. In: Grossmann,W., Pflug,
![Page 803: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/803.jpg)
G., Vincze, I., Wertz, W. (eds.) Proceedings of the 4th Pannonian Symposium on Mathematical
![Page 804: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/804.jpg)
Statistics, pp. 251–267 (1983)
![Page 805: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/805.jpg)
30. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Phila-
![Page 806: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/806.jpg)
delphia (1992)
![Page 807: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/807.jpg)
31. Nix, A.E., Vose, M.D.: Modeling genetic algorithms with Markov chains. Ann. Mathemat. Arti.
![Page 808: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/808.jpg)
Intell. 5(1), 79–88 (1992)
![Page 809: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/809.jpg)
32. Pardalos, P.M.,Romeijn,H.E. (eds.):Handbook ofGlobalOptimization, vol. 2.KluwerAcademic
![Page 810: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/810.jpg)
Publishers, Boston (2002)
![Page 811: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/811.jpg)
33. Press, W.H., Teukolsky, S.A.: Quasi- (that is, sub-) random numbers. Comput. Phys. 3(6), 76–79
![Page 812: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/812.jpg)
34. Ripley, B.D.: Spatial Statistics. John Wiley & Sons, New York (1981)
![Page 813: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/813.jpg)
35. Snyder, W.C.: Accuracy estimation for quasi-Monte Carlo simulations. Mathemat Comput. Sim-
![Page 814: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/814.jpg)
ulat. 54(1–3), 131–143 (2000)
![Page 815: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/815.jpg)
36. Sobol’, I.M.: On quasi-Monte Carlo integrations. Mathemat. Comput. Simulat. 47(2–5), 103–112
37. Sobol’, I.M., Bakin, S.G.: On the crude multidimensional search. J. Computat. Appl. Mathemat.
![Page 816: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/816.jpg)
56(3), 283–293 (1994)
![Page 817: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/817.jpg)
J Glob Optim (2007) 37:405–436 436
2
![Page 818: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/818.jpg)
38. Sobol, I.M., Tutunnikov,A.V.:A varience reducingmultiplier forMonteCarlo integrations.Math-
![Page 819: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/819.jpg)
emat. Comput. Model. 23(8–9), 87–96 (1996)
![Page 820: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/820.jpg)
39. Tang,H.-C.:Using an adaptive geneticalgorithmwith reversals to find good second-ordermultiple
![Page 821: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/821.jpg)
recursive random number generators. Mathemat. Methods Operat. Res. 57(1), 41–48 (2003)
![Page 822: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/822.jpg)
40. Testproblems for globaloptimization. http://www.imm.dtu.dk/~km/GlobOpt/testex/ , June (2004)
![Page 823: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/823.jpg)
41. Test problems in R . http://iridia.ulb.ac.be/~aroli/ICEO/Functions/Functions.html , June (2004)
![Page 824: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/824.jpg)
42. Trafalis, B., Kasap, S.: A novel metaheuristics approach for continuous global optimization.
![Page 825: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/825.jpg)
J. Global Optimizat. 23(2), 171–190 (2002)
![Page 826: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/826.jpg)
43. Tuffin, B.: On the use of low discrepancy sequences inMonte Carlo methods.Monte CarloMeth-
![Page 827: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/827.jpg)
ods Appl. 2(4), 295–320 (1996)
![Page 828: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/828.jpg)
44. Weisstein, E.W.: “circle packing”. From MathWorld–A Wolfram Web Resource. http://math-
![Page 829: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/829.jpg)
world.wolfram.com/CirclePacking.html, September (2004)
![Page 830: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/830.jpg)
45. Wikramaratna, R.S.: ACORN—a new method for generating sequences of uniformly distributed
![Page 831: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/831.jpg)
pseudo-random numbers. J. Computat. Phys. 83(1), 16–31 (1989)
![Page 832: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/832.jpg)
46. Wright, A.H., Zhao, Y.: Markov chain models of genetic algorithms. In: Banzhaf, W., Daida, J.,
![Page 833: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/833.jpg)
Eiben, A.E., Garzon, M.H., Honavar, V., Jakiela, M., Smith, R.E. (eds.) GECCO-99: Proceed- ings of the Genetic and Evolutionary Computation Conference, pp. 734–741.Morgan Kauffman,
![Page 834: genetic_POPULATIONS.doc](https://reader035.fdocuments.us/reader035/viewer/2022070301/5466ae86b4af9f583f8b545b/html5/thumbnails/834.jpg)
SanMateo, California (1999)